Research ArticleA General 3D Nonstationary Vehicle-to-Vehicle Channel ModelAllowing 3D Arbitrary Trajectory and 3D-Shaped Antenna Array

Qiuming Zhu 1 Weidong Li 1 Ying Yang1 Dazhuan Xu 1 Weizhi Zhong 2

and Xiaomin Chen 1

1Key Laboratory of Dynamic Cognitive System of Electromagnetic Spectrum SpaceCollege of Electronic and Information Engineering Nanjing University of Aeronautics and Astronautics Nanjing 211106 China2Key Laboratory of Dynamic Cognitive System of Electromagnetic Spectrum Space College of AstronauticsNanjing University of Aeronautics and Astronautics Nanjing 211106 China

Correspondence should be addressed to Qiuming Zhu zhuqiumingnuaaeducn

Received 1 May 2019 Accepted 25 August 2019 Published 20 October 2019

Guest Editor Danping He

Copyright copy 2019 Qiuming Zhu et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

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

1 Introduction

Vehicle-to-vehicle (V2V) communications can improve thesafety of life and property by collecting and exchangingenvironment information under complex traffic [1]Meanwhile multiple-input multiple-output (MIMO) tech-nologies can expand the capacity and improve the com-munication efficiency It is promising to apply MIMOtechnologies to V2V communications To provide reason-able references for developing analyzing and testing V2Vcommunication systems a general accurate and easy-to-usechannel model is required Moreover the movements in-sufficient antenna space and lack of rich scattering should beconsidered in channel modeling and statistical propertyanalysis [2ndash7]

e wide-sense stationary (WSS) assumption wasadopted in the traditional geometry-based stochastic models(GBSMs) [5 8] Nevertheless the authors in [9 10] foundthat the WSS was only suitable for short distances betweentransmitters and receivers based on the measured dataSeveral GBSMs for nonstationary V2V channels are pro-posed in the literature [11ndash25] Among them 2D non-stationary V2V GBSMs with fixed clusters [11 12] ormoving scatterers [13] were studied and the statisticalproperties ie the temporal correlation function (TCF)spatial correlation function (SCF) and Doppler powerspectrum density (DPSD) were also analyzed e authorsin [14] proposed a 2D V2V channel model with randommovement scatterers e aforementioned 2D channelmodels were only considered on the horizontal plane Under

HindawiInternational Journal of Antennas and PropagationVolume 2019 Article ID 8708762 12 pageshttpsdoiorg10115520198708762

the realistic scenarios the scatterers eg vehicles pedes-trian and infrastructures may distribute or move in the 3Dspace Meanwhile the authors in [7] have proved that the 3Dchannel model is more accurate than the 2D ones forevaluating the system performance By extending the scat-terers distributed on the surface of 3D regular shapes ie ahemisphere [15] two cylinders [16 17] two spheres [18] anelliptic-cylinder [19] a rectangular tunnel [20] several 3Dnonstationary V2V channel models were proposed recentlyIn [21] the authors proposed a 3D irregular-shaped GBSMfor nonstationary V2V channels Some 3D cluster-basednonstationary V2V channel models can also been addressedin [22ndash25] However the output phases related to theDoppler frequencies in [11ndash25] were inaccurate comparedwith the theoretical results [26 27] e authors in [28 29]modified the phase item to overcome this shortcoming butit is only suitable for 2D scattering environments Amodified 3D channel model with accurate Doppler fre-quency can be addressed in [30]

Note that most of the existing V2V channel models[11ndash26] only took the fixed velocities of the mobile trans-mitter (MT) and mobile receiver (MR) into accountHowever under realistic traffic environments the velocitiesof MT and MR could be time-variant Although the time-variant velocities were considered in [29 31ndash33] theirtrajectories were 2D Meanwhile the models in the literature[11ndash33] only considered 2D or even 1D antenna arrays forsimplicity With the development of antenna technologies3D-shaped antenna arrays begin to be used in MIMOcommunication systems Very recently a general 3D non-stationary GBSM between base station and MT with 3Darbitrary trajectories and 3D antenna arrays was proposed in[34] is idea was adopted by [35] to model the 3D V2Vchannels allowingMTandMRwith 3D arbitrary trajectoriesHowever the details of channel parameter computation andtheoretical analyses of SCF TCF and DPSD were lackedis paper aims at filling these gapsemajor contributionsand novelties are summarized as follows

(1) Combining the GBSM and twin-cluster approach[36] a general 3DV2V channel model was proposede 3D antenna array and 3D arbitrary trajectory ofeach terminal are allowed under the 3D scatteringenvironment which guarantees the proposed modelis more general and has the nonstationary aspect

(2) e upgraded computation procedure of channelparameters for the proposed model such as thenumber of paths path delays and path powers aredeveloped and analyzed in detail

(3) e theoretical closed-form expressions of statisticalproperties ie the SCFs TCFs and DPSDs areinvestigated and verified by the simulation methodand measurement data

is paper is structured as follows Section 2 presents a3D general V2V channel model characterized by non-stationary aspect and generalization e channel pa-rameter updating algorithms are given in Section 3 InSection 4 the theoretical SCFs TCFs and DPSDs of the

proposed model are derived in detail Section 5 shows andcompares the simulated results with the derived ones andmeasured data Finally the conclusions are drawn inSection 6

2 3D Nonstationary GBSM for V2V Channels

Figure 1 shows a typical 3D V2V communication systembetween the MT and MR which is characterized by 3D ar-bitrary trajectories and 3D antenna arrayseMTandMR areconfigured with S and U antennas respectively In the figurethe coordinate systems at the MT and MR with the centers ofcorresponding terminals are named as xyz and 1113957x1113957y1113957z re-spectively Suppose that the travel directions of theMTandMRat the initial time correspond to x-axisrsquos direction respectivelyDue to the movement and rotation of two coordinate systemsthe antenna element of MTor MR is 3D and time-variant andcan be expressed by dMT

s (t) [dMTsx (t) dMT

sy (t) dMTsz (t)]T or

dMRu (t) [dMR

u1113957x (t) du 1113957yMR(t) dMRu1113957z (t)]T Figure 1 shows that

between the MT and MR there are many propagation pathsand subpaths (or rays) e 3D location of the first cluster SMT

n

or the last cluster SMRn affects the corresponding elevation

angles such as elevation angle of departure (EAoD) θMTnm(t)

and elevation angle of arrival (EAoA) θMRnm(t) and azimuth

angles such as the azimuth angle of departure (AAoD) ϕMTnm(t)

and the azimuth angle of arrival (AAoA) ϕMRnm(t) In addition

by adopting twin-cluster approach the rest clusters can beviewed as a virtual link [36] In the figure vi(t) and vSi

n (t)i isin MTMR represent time-variant velocities of i and Si

nrespectively Table 1 shows the detailed definitions of channelparameters

Under the V2V communication scenario of Figure 1 theV2V MIMO channel can be modeled as [30]

Ht τ

h11t τ h12t τ middot middot middot h1St τ

h21t τ hust τ middot middot middot h2St τ

⋮ ⋮ ⋱ ⋮

hU1t τ hU2t τ middot middot middot hUSt τ

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)

where hus(t τ) denotes the complex channel impulse re-sponse (CIR) between the transmitting antennas(s 1 2 S) and receiving antenna u(u 1 2 U)In this paper we modify the model of hus(t τ) in [30] andexpress it as

hus(t τ)≜ΠT0(t) 1113944

N(t)

n1

Pn(t)

11139691113957husn(t)δ τ minus τn(t)( 1113857 (2)

where ΠT0(t) is a rectangular window function

ΠT0(t)≜

1 0le tleT0

0 otherwise1113896 (3)

and it is introduced to limit the length of CIR and thus thelarge-scale variations can be negligible within the time in-terval N(t) represents the valid path number and is relatedto the path delay τn(t) path power Pn(t) and normalizedcoefficient 1113957husn(t) which is expressed as

2 International Journal of Antennas and Propagation

1113957husn(t) limM⟶infin

1M

1113970

1113944

M

m1ej ΦDnm(t)+ΦLusnm(t)+ΦInm( ) (4)

where M denotes the subpath number ΦInm stands for therandom initial phase distributing over [0 2π) uniformlyand ΦDnm(t) means the phase somehow affected by variantDoppler frequency Here we model ΦDnm(t) as

ΦDnm(t) k 1113946t

0vMTSMT

n tprime( 1113857 middot 1113957sMTnm tprime( 1113857 + vS

MRn MR

tprime( 1113857

middot 1113957sMRnm tprime( 1113857dtprime

(5)

where k 2πfcc denotes the wave number with fc and crepresenting the carrier frequency and light speed re-spectively viSi

n (t) i isin MTMR denotes the relative ve-locity between i and Si

n e departure or arrival angle unit

vector of the mth subpath within the nth path 1113957sinm(t) is

defined as

sinm(t)

cos θinm(t)cos ϕi

nm(t)

cos θinm(t)sin ϕi

nm(t)

sin θinm(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (6)

In (4) ΦLusnm(t) denotes the offset phase caused by themovements of terminals

ΦLusnm(t) k middot 1113957sMTnm(t)1113872 1113873

Tmiddot RMT

v (t) middot dMTt0s + k middot 1113957sMT

nm(t)1113872 1113873T

middot RMTv (t) middot dMTt0

u

(7)where [middot]T means transpose operation and dMTt0

s representssth antenna position at initial time instant Similarly dMRt0

u

θvMR(t)

x~x~

x~

x~vMR(t)

vMR(t)

vMR(t)

vMR(t)

z~ z~

z~

z~

dsMR(t)

dsMR(t)

ϕvMR(t)ϕnm

MR(t)

ϕnmMR(t)

MR(t)θnm

MR(t)θr

MR(t)θnm

y~

y~y~

y~

SnMR

SnMT

vSnMR

(t)

vSnMR

(t)

zz

z

z

y

y

yy

xxx

ϕvMR(t)

Figure 1 Typical 3D V2V communication system

Table 1 Definitions of channel parameters

N(t) Valid path numberPn(t) τn(t) e nth path power and delayM e number of subpaths

dMTs (t) dMR

u (t)Location vectors of transmitter antenna element s and

receiver antenna element u respectively

Li(t) LSin (t)

Location vectors of i and Sin i isin MTMR

respectively

vi(t) vSin (t)

Velocity vectors of i and Sin i isin MTMR

respectively||vi(t)|| ||vS

in (t)|| Speeds of i and Si

n i isin MTMR respectively

ϕiv(t)ϕSi

nv (t)

Traveling azimuth angles of i and Sin i isin MTMR

respectively

θiv(t) θSi

nv (t)

Traveling elevation angles of i and Sin i isin MTMR

respectively

1113957sinm(t)

Spherical unit vector of i along ray m of cluster ni isin MTMR

ϕMTnm(t) θMT

nm(t)Azimuth and elevation departure angles along ray m

of cluster n respectively

ϕMRnm(t) θMR

nm(t)Azimuth and elevation arrival angles along ray m of

cluster n respectively

International Journal of Antennas and Propagation 3

represents the corresponding antenna position at initialcorresponding time Note that Ri

v(t) is a rotation matrix dueto the 3D arbitrary trajectory and it can be written as

Riv(t)

cos θiv(t)cos ϕi

v(t) minus sin ϕiv(t) minus sin θi

v(t)cos ϕiv(t)

cos θiv(t)sin ϕi

v(t) cosϕiv(t) minus sin θi

v(t)sin ϕiv(t)

sin θiv(t) 0 cos θi

v(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)

3 Time-Variant Channel Parameters

31 Path Number e valid path number has been provedto be time-variant under V2V scenarios by the measureddata in [37] due to themoving clusters and two terminals Inother words the old disappearing clusters and newappearing ones are random We model these with a Markovprocess [38] λG is used to denote the birth rate and λR

denotes the death rate e probability of each pathremaining from t to t + Δt is calculated as [19]

Pr(tΔt) eminus λR middotPc vSMT(t)

+ vSMR

(t)

( 1113857Δt

middot e minus λR vMR(t)minus vMT(t) Δt( )

(9)

where Pc means the moving percentage and ||vSi

(t)||i isin MTMR represents clusters average speed Nnew(t) isintroduced to describe newly generated path number andcalculated as

E Nnew(t)1113864 1113865 λG

λR

1 minus Pr(tΔt)( 1113857 (10)

Combining (9) with (10) the averaged path number canbe expressed as

E N(t) N(t minus Δt)Pr(tΔt) + E Nnew(t)1113864 1113865 λG

λR

(11)

32Delays andPowers For the nth valid path the total delayat time instant t consists of the first and last bounce delaysand virtual link delay and it is a function of total distance as

τn(t) LMT(t) minus LS

MTn (t)

+ LMR(t) minus LSMRn (t)

c+ 1113957τn(t)

(12)

where Li(t) and LSin (t) i isin MTMR mean the in-

stantaneous locations of i and Sin and can be updated from

the values of previous time instant by

Li(t) Li

(t minus Δt) + vi(t minus Δt)Δt

LSin (t) LS

in (t minus Δt) + vS

in (t minus Δt)Δt

(13)

where 1113957τn(t) denotes the equivalent delay of virtual link and itcan be updated by the first-order filtering method in [26] asfollows

1113957τn(t) 1113957τn(t minus Δt)eminus Δtτdec( ) + 1 minus eminus Δtτdec( )1113874 1113875X (14)

where X sim U[LMT(t) minus LMR(t)c τmax] in which τmaxdenotes the maximum delay and τdec denotes the decor-relation speed of time-variant delays Based on the mea-surement-based method we can get the averaged power as

Pnprime(t) eminus τn(t) 1minus rDSrDSσDS( ) times 10minus ξn10 (15)

where ξn rDS and σDS mean the shadowing degree delayfactor and delay spread respectively which are all relatedwith the environments Finally all the normalized power canbe denoted as

Pn(t) Pnprime(t)

1113936N(t)n1 Pnprime(t)

(16)

33 Time-Variant Angles Measurements in [39] revealedthat the azimuth and elevation angles at two terminals arenot independent and cannot be depicted by two independentdistributions such as Gaussian or Laplacian In this paper wetake the 3D von MisesndashFisher (VMF) distribution to de-scribe these two joint angles [40] e probability densityfunction (PDF) of VMF distribution can be expressed as [39]

p(ϕ θ) κ cos θeκ(cos θ cos θ cos(ϕminus ϕ)+sin θ sin θ)

4π sinh(κ)

minus π le ϕle π minusπ2le θle

π2

(17)

where ϕ and θ represent the mean values of azimuth andelevation angles respectively and κ is the shape factor andcan be extracted from the measurement data

As long as κ is determined the time-variant angle ofdeparture (AoD) and angle of arrival (AoA) can be obtained byϕ(t) and θ(t) respectively which can be expressed as follows

θi

n(t) arcsinLS

in

z (t) minus Liz(t)

LSin (t) minus Lit)

⎛⎝ ⎞⎠

ϕi

n(t)

arccosLS

in

x (t) minus Lix(t)

LSin (t) minus Li(t)

1113874 1113875cos θi

n(t)1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

LSin

y (t) minus Liy(t)ge 0

minus arccosLS

in

x (t) minus Lix(t)

LSin (t) minus Li(t)

1113874 1113875cos θi

n(t)1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

LSin

y (t) minus Liy(t)lt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

where i isin MTMR Note that κ almost unchanged duringthe short simulation time period Figure 2 gives an exampleof VMF distributed angles of AoA (or AoD) on the unitsphere with different parameters It also reveals that with the

4 International Journal of Antennas and Propagation

increase in shape factor κ the shape of distribution becomesmore concentrated

4 Time-Variant Statistical Properties

e normalized spatial-temporal correlation function(STCF) between two different subchannels can be defined by[16 19 25]

1113957Ru2 s2n

u1 s1n(tΔtΔd) E 1113957hu1 s1n(t)1113957h

lowastu2 s2 n(t + Δt)1113960 1113961

E 1113957hu1 s1 n(t)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877E 1113957hu2 s2 n(t + Δt)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(19)

where E[middot] represents the expectation function ( )lowast iscomplex conjugate and Δt means time lag Moreover Δd

ΔdMTΔdMR1113966 1113967 denotes the space lag and consists of ΔdMT

and ΔdMR which are the corresponding space lags at twoterminals respectively By substituting our proposed

channel model (4) into (19) the expression of STCF can beobtained as

1113957Ru2 s2 n

u1 s1 n(tΔtΔd) limM⟶infin

1

M

1113970

1113944

M

m1ej ΦDu2 s2 nm(t+Δt)+ΦLu2 s2 nm(t+Δt)+ΦInm1113872 1113873eminus j ΦDu1 s1 nm(t)+ΦLu1 s1 nm(t)+ΦInm1113872 1113873

1113946111394611139461113946

θMTn ϕMT

n θMRn ϕMR

n

p ϕMTn (t + Δt) θMT

n (t + Δt)1113872 1113873

1113969

p ϕMRn (t + Δt) θMR

n (t + Δt)1113872 1113873

1113969

middot

p ϕMRn (t) θMR

n (t)1113872 1113873

1113969

middot

p ϕMTn (t) θMT

n (t)1113872 1113873

1113969

ej ΦDu2 s2 n(t+Δt)minus ΦDu1 s1 n(t)+ΦLu2 s2 n(t+Δt)minus ΦLu1 s1 n(t)1113872 1113873dθMT

n dϕMTn dθMR

n dϕMRn

(20)

41 Time-Variant SCFs e SCFs is reduced from STCFwhen the time lagΔt equals to zeroen it can be expressed as

1113957Ru2s2 n

u1s1 n(tΔd) 1113946111394611139461113946

θMTn ϕMT

n θMRn ϕMR

n p ϕMT

n (t) θMTn (t)1113872 1113873p ϕMR

n (t) θMRn (t)1113872 1113873 middot ej ΦLu2 s2 n(t)minus ΦLu1 s1 n(t)1113872 1113873

dθMTn dϕMT

n dθMRn dϕMR

n

(21)

Since the clusters SMTn and SMR

n are independent we canrewrite (21) as

1113957Ru2 s2 n

u1 s1 n(tΔd) 1113957RMTn tΔdMT

1113872 1113873 middot 1113957RMRn tΔdMR

1113872 1113873 (22)

where 1113957Ri

n(tΔdi) i isin MTMR denotes the normalizedSCF at the MT or MR By substituting (7) into (21) thefollowing equation can be obtained

1113957Ri

n tΔdi1113872 1113873 1113946

π

minus π1113946π

minus πejk 1113957s i

n (t)( 1113857T

middotRiV(t)middotΔdit0

middot pϕinm(t)θi

nm(t) ϕin(t) θi

n(t)1113872 1113873dϕindθi

n

(23)

It is shown in (23) that not only the angle of horizontalplane ϕi

n(t) but also the one of vertical plane θin(t) affects the

result of (23) us traditional 2D models cannot obtainaccurate correlation functions in 3D scattering environ-ments By setting θi

n(t) θi

n(t) + ζ and ϕin(t) ϕi

n(t) + υand substituting them into (23) it yields

1113957Ri

n tΔdi1113872 1113873 1113946

1113946

ζ υ

κ cos θi

n(t) + ζ1113874 1113875AiBi

4π sinh(κ)dζdυ

(24)

where

Ai

eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)cos(υ)+sin θi

n(t)+ζ( 1113857sin θi

n(t)( 1113857 (25)

Bi

ejk1113957s ϕi

n(t)+υθi

n(t)+ζ( 1113857RiV(t)Δdit0

(26)

ndash1ndash05ndash05

05

005

1

Xndash1

0

1ndash1

ndash05

0

05

1

Y

Z

κ = 395

θ = ndash6degϕ = 130degndashndash

κ = 2773

θ = 0degϕ = 10degndashndash

κ = 65

θ = 6degϕ = 240degndashndash

Figure 2 e VMF distributed angles on the unit sphere

International Journal of Antennas and Propagation 5

e measurement data demonstrate that the anglespread is usually narrow under V2V communication sce-narios With this assumption we have cos(ζ) asymp cos(υ) asymp 1sin(ζ) asymp ζ sin(υ) asymp υ κcos(υ) asymp κ(1 minus υ22) and κcos(ζ)

asymp κ(1 minus ζ22) Consequently we can rewrite (25) and (26) as

Ai asymp eκ 1minus ζ22minus 2 cos2 θ

i

n(t)( 1113857(υ2)2( 1113857 (27)

Bi asymp ejk 1113957s ϕ

i

n(t)θi

n(t)( 1113857+Ci(t)ζ+Ei(t)υ( 1113857RiV(t)Δdit0

(28)

where

Ci(t) minus sin θ

i

n(t)cosϕi

n(t) minus sin θi

n(t)sin ϕi

n(t) cos θi

n(t)1113876 1113877

(29)

Di(t) minus sinϕi

n(t) cosϕi

n(t) 01113876 1113877 (30)

By substituting (27)ndash(30) into (24) and using Eulerformula we can obtain the following equation

1113957Ri

n tΔdi1113872 1113873 asymp

ejk1113957s ϕi

n(t)θi

n(t)( 1113857RiV(t)Δdit0+κcos( θ

i

n(t)1113857

κ1minus m2( )(2π)m2Im2minus 1(κ)

middot 1113946Δiϕ

minus Δiϕ

eminus Ei(t)υ2 cos kDi(t)Ri

V(t)Δdit0υ1113872 1113873dυ

middot 1113946Δiθ

minus Δiθ

eminus κζ22cos kCi(t)Ri

V(t)Δdit0ζ1113872 1113873dζ

(31)

where Ei(t) κ cos2(θi

n(t))2 and Δiϕ and Δi

θ are the anglespreads of AAoA and EAoA With the help of integrationformula

1113946c

minus ceminus ax2

cos(bx)dx j

π

radiceminus b24a(erfi(b minus 2jac)2

a

radic) minus erfi(b + 2jac2

a

radic)

2a

radic (32)

e approximate closed-form result of SCF is as follows

1113957Ri

n tΔdi1113872 1113873 asymp

minus Im erfi Fi(t)( 1113857( 1113857Im erfi Gi(t)( 1113857( 1113857

8 sinh(κ)

middot ejk1113957s ϕi

n(t)θi

n(t)( 1113857RiV(t)Δdit0

middot eminus kRiV(t)Δdit0( )

2 Di(t)2κ

radic( )

2+ Ci(t)

2κ

radic( )

2( 1113857+κ

(33)

where erfi(middot) is the imaginary error function

Fi(t)

kDi(t)RiV(t)Δdit0 minus jκcos2 θ

i

n(t)1113874 1113875Δiϕ

2κcos2 θi

n(t)1113874 1113875

1113970 (34)

Gi(t)

kCi(t)RiV(t)Δdit0 minus jκΔi

θ2κ

radic (35)

1113957Ru1 s1 n

u1 s1 n(tΔt) 1113946111394611139461113946

θMTn ϕMT

n θMRn ϕMR

n

p ϕMTn (t + Δt) θMT

n (t + Δt)1113872 1113873

1113969

p ϕMRn (t + Δt) θMR

n (t + Δt)1113872 1113873

1113969

p ϕMRn (t) θMR

n (t)1113872 1113873

1113969

middot

p ϕMTn (t) θMT

n (t)1113872 1113873

1113969

ej ΦDu1 s1 n(t)minus ΦDu1 s1 n(t+Δt)+ΦLu1 s1 n(t)minus ΦLu1 s1 n(t+Δt)1113872 1113873dθMT

n dϕMTn dθMR

n dϕMRn

(36)

1113957Ri

n(tΔt) Bϕi

nθin

p ϕin(t) θi

n(t)1113872 1113873

1113969

p ϕin(t + Δt) θi

n(t + Δt)1113872 1113873

1113969

eminus jk 1113946

t+Δt

tviSi

n tprime( 1113857 middot sinm tprime( 1113857dtprime

dϕindθi

n (37)

6 International Journal of Antennas and Propagation

1113957Ri

n(tΔt) κ

4π sinh(κ)1113946Δθ

minus Δθ1113946Δϕ

minus Δϕ

cos θi

n(t) + ζ1113874 1113875eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)( 1113857cos(υ)+sin θi

n(t)+ζ( 1113857sinθi

n(t)( 11138571113970

middot

cos θi

n(t + Δt) + ζ1113874 1113875eκ cos θi

n(t+Δt)+ζ( 1113857cosθi

n(t+Δt)cos(υ)+sin θi

n(t+Δt)+ζ( 1113857sinθi

n(t+Δt)( 11138571113970

middot eminus jk1113938t+Δt

tviSi

n (t)

cos ϕ

iSin

v (t)minus ϕi

n(t)+υ( 11138571113872 1113873cos θi

n(t)+ζ( 1113857cosθiSi

nv (t)+sin θ

i

n(t)+ζ( 1113857sinθiSi

nv (t)1113960 1113961dt

dυdζ

(38)

1113957Ri

n(tΔt) κ

4π sinh(κ)1113946Δθ

minus Δθ1113946Δϕ

minus Δϕ

cos θi

n(t) + ζ1113874 1113875eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)( 1113857cos(υ)+sin θi

n(t)+ζ( 1113857sinθi

n(t)( 11138571113970

middot

cos θi

n(t + Δt) + ζ1113874 1113875eκ cos θi

n(t+Δt)+ζ( 1113857cosθi

n(t+Δt)cos(υ)+sin θi

n(t+Δt)+ζ( 1113857sinθi

n(t+Δt)( 11138571113970

middot eminus jk viSin (t)

cos Ai(tΔtυζ)( )Di(tΔt)+cos Bi(tΔtυζ)( )Ei(tΔt)+sin Ci(tΔtζ)( )Fi(tΔt)[ ]dυdζ

(39)

Finally by substituting (33)ndash(35) into (22) the finalresult of SCF in our proposed model can be obtained Due totime-variant communication environments SCF is alsotime-dependent

42 Time-Variant TCFs e TCF is reduced from STCFwhen Δd is set to be zero en TCF can be derived as(36) It also equals to the product of TCFs at the MT andMR 1113957R

i

n(tΔt) i isin MTMR when the clusters SMTn and

SMRn are independent For simplicity we assumed that theantenna array is placed at the origin of the coordinate

system By substituting (5) and (7) into (36) it can beobtained as (37)

By setting θin(t) θ

i

n(t) + ζ and ϕin(t) ϕi

n(t) + υ andsubstituting them into (37) it yields (38) where ϕiSi

nv (t) and

θiSin

v (t) mean the azimuth and elevation angles of the relativevelocity between i and cluster Si

n respectively It is rea-sonable to assume that the elevation AoDs and azimuthAoDs change linearly during the short time interval asϕi

n(t) k1t + b1 θi

n(t) k2t + b2 where k1 (ϕi

n(t + Δtmax) minus ϕi

n(t))Δtmax k2 (θi

n(t + Δtmax) minus θi

n(t))Δtmaxb1 ϕi

n(t) minus k1t and b2 θi

n(t) minus k2t By integrating theDoppler frequency we can rewrite (38) as (39) where

Ai(tΔt υ ζ)

t + Δt2

1113874 1113875 minus k1 + k2( 1113857 + ϕiSin

v (t) minus b1 + b2 minus υ + ζ

Bi(tΔt υ ζ)

t + Δt2

1113874 1113875 minus k1 minus k2( 1113857 + ϕiSin

v (t) minus b1 minus b2 minus υ minus ζ

Ci(tΔt ζ) k2

t + Δt2

1113874 1113875 + b2 + ζ

Di(tΔt) cos θiSi

n

v (t)sinΔt minus k1 + k2( 11138572( 1113857

minus k1 + k2( 11138571113888 1113889

Ei(tΔt) cos θiSi

n

v (t)sinΔt k1 + k2( 11138572( 1113857

k1 + k2( 11138571113888 1113889

Fi(tΔt) 2sin θiSi

n

v (t)sink2Δt2( 1113857

k21113888 1113889

(40)

1113957Ri

n(tΔt) asympκeκ

4π sinh(κ)

cos θi

n(t)1113874 1113875cos θi

n(t + Δt)1113874 1113875

1113970

eminus jkGi(tΔt)

middot 1113946Δθ

minus Δθeminus κζ22cos kI

i(tΔt)ζ1113872 1113873dζ middot 1113946

Δϕ

minus Δϕeminus Ji(tΔt)υ2 cos kH

i(tΔt)υ1113872 1113873dυ

(41)

International Journal of Antennas and Propagation 7

Measurement data have revealed that small angle spreadsexist under some scenarios In other words κ is usually bigor ζ and υ are small Holding this condition we have

cos(υ minus ζ) asymp 1 cos(υ + ζ) asymp 1 sin(υ minus ζ) asymp υ minus ζ sin(υ + ζ)

asymp υ + ζ cos(ζ) asymp 1 cos(υ) asymp 1 sin(ζ) asymp ζ and sin(υ) asymp υus we can approximate (39) as (41) where

Gi(tΔt) viSi

n (t)

middot Di(tΔt)cos A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)cos B

i(tΔt)1113872 1113873 + viSi

n (t)

middot Fi(tΔt)sin C

i(tΔt)1113872 1113873

Hi(tΔt) viSi

n (t)

middot Di(tΔt)sin A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)sin B

i(tΔt)1113872 1113873

Ii(tΔt) minus viSi

n (t)

middot Di(tΔt)sin A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)sin B

i(tΔt)1113872 1113873 + viSi

n (t)

middot Fi(tΔt)cos C

i(tΔt)1113872 1113873

Ji(tΔt)

κ cos2 θi

n(t)1113874 1113875 + cos2 θi

n(t + Δt)1113874 11138751113874 1113875

4

(42)

1113957Ri

n(tΔt) minusκ

8 sinh(κ)

cos θi

n(t)1113874 1113875cos θi

n(t + Δt)1113874 1113875

1113970

eκminus k2Ii2(tΔt)2κ( 1113857minus k2Hi2(tΔt)4Ji(tΔt)( 1113857minus jkGi(tΔt)

middotIm erfi kHi(tΔt) minus 2jJi(tΔt)Δϕ2

Ji(tΔt)

11139681113872 11138731113872 1113873

Ji(tΔt)

1113968Im erfi kIi(tΔt) minus jκΔθ

2κ

radic( 1113857( 1113857( 1113857

2κ

radic

(43)

where Ai(tΔt) Bi(tΔt) and Ci(tΔt) are the value ofAi(tΔt υ ζ) Bi(tΔt υ ζ) and Ci(tΔt ζ) when υ ζ 0respectively Using the integration formula in (32) theclosed-form expression of (41) can be expressed as (43) Onthis basis we can obtain the final TCF expression of theproposed model

43 Time-Variant DPSDs e DPSD is the Fourier trans-form of TCF 1113957Rusn(tΔt) For the nonstationary aspect of ourproposed model the DPSD can be calculated with short-time Fourier transform as

Sn(f t) 1113946infin

minus infin1113957Rusn(tΔt) eminus j2πfΔt](t minus Δt)dΔt (44)

where window function ](t minus Δt) lasting time is shorter thanthe stationary interval (about several milliseconds [10])

5 Simulation Results and Validation

Firstly in order to evaluate the accuracy of our derived SCFand TCF we calculate the maximum difference between theapproximated and numerical integral results overϕi

n isin [minus 180deg 180deg] and θi

n isin [minus 10deg 10deg] Figure 3 shows theresults of maximum absolute error with different κ nor-malized space lags and time lags As we can see that themaximum absolute error of SCF is less than 0025 when κ ismore than 50 and dλ is less than 3 and it increases as κdecreases or dλ increases Meanwhile the absolute error ofTCF is less than 002 when κ is more than 50 and the time lagis less than 005 s and it increases as κ decreases or the timelag increases Overall the maximum error is acceptable andthus the derived expressions can be used to calculate theSCF and TCF efficiently

Secondly in order to verify the generality of the proposedchannel model ree V2V communication scenarios withdifferent trajectories are selected and given in Figure 4 In thefigure theMTtravels straightly with the fixed velocity and theMR travels in three different velocities with the same de-parture and arrival points Path I has fixed speed and traveldirection and Path II allows speed variation with 2D directionmovement Furthermore Path III has 3D variations in bothvertical and horizontal directions For demonstration pur-pose the distribution of clusters is uniform and we adoptVMF distribution to depict AoAs and AoDs and set κ as395 [39] With the referred data in [41] we assume thatthe speeds of cluster are Gaussian-distributed with 1 kmhmean value and 01 variance Furthermore the carrier fre-quency fc 24GHz ϕS

MTnv ϕS

MRnv θS

MTnv and θS

MRnv are all uni-

formly distributed e former pair distributes over 0 and 2πbut the later one only distributes over minus π36 and π36

It should be mentioned that the V2V channel models in[11ndash26 28 30] only considered fixed velocities of two ter-minals like Path I while the models in [31ndash33] allowed for2D curve trajectories like Path II e absolute values of thetheoretical SCFs of different models including the modelconsidering Path I in [30] and the one focusing on Path II in[32] are simulated and compared in Figure 5 e goodagreement of SCFs between our proposed and other modelsindicates the generalization and compatibility of our pro-posed model In particular the models in [30 32] can beviewed as two special applications in straight or curvedtrajectories

irdly in order to verify the correctness of statisticalproperties of our proposed model under 3D trajectories likePath III the theoretical approximated and simulated valuesof SCFs and TCFs are shown in Figure 6 e x-axis ofFigure 6(a) represents normalized space between antennas

8 International Journal of Antennas and Propagation

320

140

005

120

001

1100 80

0015

60

002

40 200

Max

imum

abso

lute

erro

r

Shape factor ()Normalized antenna

spacing (Δdλ)

(a)

00025

005 200150

10050

Shape factor (κ)Time lag (∆t (s))

0

0005

001

0015

002

Max

imum

abso

lute

erro

r

(b)

Figure 3 Maximum absolute errors between the theoretical and approximate (a) SCFs and (b) TCFs

Start

EndEnd

Start

Path I (1D straight)

Path III (3D arbitrary)

Path II (2D curve)

MTMR

50100 0 ndash50 ndash100 ndash1500

50100

Postion y(m)

Postion x(m)

Posti

on z

(m)

0

1

4

3

5

2

Figure 4 ree typical trajectories of the MR

0

02

04

06

08

1

Model in [28]Model in [30]Proposed model

t = 4st = 8s

t = 0s

Normalized antenna space (Δdλ)0 05 1 15 2

Abso

lute

val

ue o

f SCF

(a)

02

04

06

08

1

Model in [30]Proposed model

00

05 1 15 2

t = 4s

t = 8s

t = 0s

Abso

lute

val

ue o

f SCF

Normalized antenna space (Δdλ)

(b)

Figure 5 Comparison of absolute SCFs for (a) Path I and (b) Path II

International Journal of Antennas and Propagation 9

which is similar at two terminals in the communicationsystem By substituting u1 s1 1 u2 s2 2 and n 1into (21) and (33) we can get the theoretical and approx-imated values of SCFs In addition the measured results in[42] are also shown in Figure 6(a) e good agreement ofSCFs shows the correctness of both the theoretical modeland derivations Similarly by substituting u s 1 and n

1 into (36) and (43) we obtain and show the theoretical andapproximated values of TCFs in Figure 6(b) e goodagreements between theoretical simulated and approxi-mated results reveal that our simulation and derivation arecorrect

Finally substituting TCFs into (44) we can get thetheoretical and simulated values of DPSDs for Path IIIwhich are also shown in Figure 7 It clearly shows thatmoving clusters and two terminals have an influence on thedrifting of DPSDs Moreover good agreements between thesimulated and theoretical results indicate that our proposedmodel is correct

6 Conclusion

is paper has proposed a general channel model charac-terized by 3D scattering environments 3D movements and

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)Measurement

0 05 1 15 2

t = 4s

t = 8s

t = 0s

Normalized antenna space (Δdλ)

Abso

lute

val

ue o

f SCF

(a)

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

0 0005 001 0015 002 0025 003 0035 004

Abso

lute

val

ue o

f TCF

Time lag (Δt)

t = 8s

(b)

Figure 6 Comparison of absolute theoretical and simulated (a) SCFs and (b) TCFs at different time instants

86

4ndash5

0

ndash1000

5

2

10

ndash500

15

0 500 01000Frequency f (HZ)

DPS

D

Time t (s)

(a)

0

5

10

15

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

t = 8s

ndash600 ndash400 ndash200 0 200 400 600Frequency f (HZ)

DPS

D

(b)

Figure 7 (a) eoretical 3D DPSD and (b) theoretical and simulated DPSDs at different time instants for Path III

10 International Journal of Antennas and Propagation

3D-shaped antenna arrays of two terminals for V2V com-munication system e upgraded algorithms for time-evolving parameters such as the number of paths pathdelays path powers and angles have been developed andanalyzed in detail Meanwhile the approximated expressionsof statistical properties including SCF TCF and DPSD havebeen derived and verified by simulations Simulated andanalyzed results show that different moving trajectoriessignificantly affect the V2V channel characteristic Mean-while the generalization and compatibility of our proposedmodel also demonstrate that some previous models with 2Dor even 1D movements have special applications e newmodel can be used to develop analyze and test realistic V2Vcommunication systems in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Key ScientificInstrument and Equipment Development Project (Grant no61827801) Aeronautical Science Foundation of China(Grant no 2017ZC52021) and Open Foundation forGraduate Innovation of NUAA (Grant no KFJJ 20180408)

References

[1] C-X Wang X Cheng and D I Laurenson ldquoVehicle-to-vehicle channel modeling andmeasurements recent advancesand future challengesrdquo IEEE Communications Magazinevol 47 no 11 pp 96ndash103 2009

[2] K Guan B Ai M Liso Nicolas et al ldquoOn the influence ofscattering from traffic signs in vehicle-to-x communicationsrdquoIEEE Transactions on Vehicular Technology vol 65 no 8pp 5835ndash5849 2016

[3] X Cheng C-X Wang B Ai and H Aggoune ldquoEnvelopelevel crossing rate and average fade duration of nonisotropicvehicle-to-vehicle ricean fading channelsrdquo IEEE Transactionson Intelligent Transportation Systems vol 15 no 1 pp 62ndash722014

[4] Q Zhu C Xue X Chen and Y Yang ldquoA newMIMO channelmodel incorporating antenna effectsrdquo Progress In Electro-magnetics Research M vol 50 pp 129ndash140 2016

[5] A G Zajic and G L Stubber ldquoSpace-time correlated mobile-to-mobile channels modelling and simulationrdquo IEEETransactions on Vehicular Technology vol 57 no 2pp 715ndash726 2008

[6] W Fan X Carreno Bautista de Lisbona F Sun J O NielsenM B Knudsen and G F Pedersen ldquoEmulating spatialcharacteristics of MIMO channels for OTA testingrdquo IEEETransactions on Antennas and Propagation vol 61 no 8pp 4306ndash4314 2013

[7] J Zhang Y Zhang Y Yu R Xu Q Zheng and P Zhangldquo3-D MIMO how much does it meet our expectationsobserved from channel measurementsrdquo IEEE Journal on

Selected Areas in Communications vol 35 no 8pp 1887ndash1903 2017

[8] J Karedal F Tufvesson N Czink et al ldquoA geometry-basedstochastic MIMO model for vehicle-to-vehicle communica-tionsrdquo IEEE Transactions onWireless Communications vol 8no 7 pp 3646ndash3657 2009

[9] A Paier T Zemen L Bernado et al ldquoNon-WSSUS vehicularchannel characterization in high-way and urban scenarios at52 GHz using the local scattering functionrdquo in Proceedings ofthe 2008 International ITG Workshop on Smart Antennaspp 9ndash15 Darmstadt Germany February 2008

[10] R He O Renaudin V-M Kolmonen et al ldquoCharacterizationof quasi-stationarity regions for vehicle-to-vehicle radiochannelsrdquo IEEE Transactions on Antennas and Propagationvol 63 no 5 pp 2237ndash2251 2015

[11] YMa L Yang andX Zheng ldquoA geometry-based non-stationaryMIMO channel model for vehicular communicationsrdquo ChinaCommunications vol 15 no 7 pp 30ndash38 2018

[12] C A Gutirrez J T Gutirrez-Mena J M Luna-RiveraD U Campos-Delgado R Velzquez and M Patzold ldquoGe-ometry-based statistical modeling of non-WSSUS mobile-to-mobile Rayleigh fading channelsrdquo IEEE Transactions onVehicular Technology vol 67 no 1 pp 362ndash377 2018

[13] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the Vehicular Technology Confer-ence pp 1ndash6 San Francisco CA USA September 2011

[14] A Borhani and M Patzold ldquoCorrelation and spectralproperties of vehicle-to-vehicle channels in the presence ofmoving scatterersrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 9 pp 4228ndash4239 2013

[15] H Jiang Z Zhang L Wu and J Dang ldquoNovel 3-D irregular-shaped geometry-based channel modeling for semi-ellipsoidvehicle-to-vehicle scattering environmentsrdquo IEEE WirelessCommunications Letters vol 7 no 5 pp 836ndash839 2018

[16] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014

[17] X Zhao X Liang S Li and B Ai ldquoTwo-cylinder and multi-ring GBSSM for realizing and modeling of vehicle-to-vehiclewideband MIMO channelsrdquo IEEE Transactions on IntelligentTransportation Systems vol 17 no 10 pp 2787ndash2799 2016

[18] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA novel 3Dnonstationary channel model based on the von Mises-Fisherscattering distributionrdquo Mobile Information Systemsvol 2016 Article ID 2161460 pp 1ndash9 2016

[19] Y Yuan C-X Wang Y He M M Alwakeel andE-H M Aggoune ldquo3D wideband non-stationary geometry-based stochastic models for non-isotropic MIMO vehicle-to-vehicle channelsrdquo IEEE Transactions on Wireless Communi-cations vol 14 no 12 pp 6883ndash6895 2015

[20] N Avazov S M R Islam D Park and K S Kwak ldquoStatisticalcharacterization of a 3-D propagation model for V2Vchannels in rectangular tunnelsrdquo IEEE Antennas andWirelessPropagation Letters vol 16 pp 2392ndash2395 2017

[21] H Jiang Z Zhang J Dang and L Wu ldquoA novel 3-D massiveMIMO channel model for vehicle-to-vehicle communicationenvironmentsrdquo IEEE Transactions on Communicationsvol 66 no 1 pp 79ndash90 2018

[22] N Avazov S M R Islam D Park and K S Kwak ldquoCluster-based non-stationary channel modeling for vehicle-to-vehiclecommunicationsrdquo IEEE Antennas and Wireless PropagationLetters vol 16 pp 1419ndash1422 2016

International Journal of Antennas and Propagation 11

[23] D Du X Zeng X Jian L Miao and H Wang ldquoree-di-mensional vehicle-to-vehicle channel modeling with multiplemoving scatterersrdquo Mobile Information Systems vol 2017Article ID 7231417 pp 1ndash14 2017

[24] X LiangW Cao and X Zhao ldquoDoppler power spectra for 3Dvehicle-to-vehicle channels with moving scatterersrdquo IEEEAccess vol 6 pp 42822ndash42828 2018

[25] X Chen Y Fang Y Sun Y Pan and W Xiang ldquoStatisticalcharacterization of novel 3D cluster-based MIMO vehicle-to-vehicle models for urban street scattering environmentsrdquoInternational Journal of Antennas and Propagation vol 2018Article ID 6742346 pp 1ndash11 2018

[26] Q Zhu H Li Y Fu et al ldquoA novel 3D non-stationary wirelessMIMO channel simulator and hardware emulatorrdquo IEEETransactions on Communications vol 66 no 9 pp 3865ndash3878 2018

[27] K Jiang X Chen Q Zhu L Chen D Xu and B Chen ldquoAnovel simulation model for nonstationary rice fading chan-nelsrdquo Wireless Communications and Mobile Computingvol 2018 Article ID 8086073 pp 1ndash9 2018

[28] M Patzold C A Gutierrez and N Youssef ldquoOn the con-sistency of non-stationary multipath fading channels withrespect to the average doppler shift and the doppler spreadrdquoin Proceedings of the IEEE Wireless Communications andNetworking Conference (WCNC) pp 1ndash6 San Francisco CAUSA 2017

[29] Q Zhu W Li C-X Wang et al ldquoTemporal correlations for anon-stationary vehicle-to-vehicle channel model allowingvelocity variationsrdquo IEEE Communications Letters vol 23no 7 pp 1280ndash1284 2019

[30] Q Zhu Y Yang X Chen et al ldquoA novel 3D non-stationaryvehicle-to-vehicle channel model and its spatial-temporalcorrelation propertiesrdquo IEEE Access vol 6 pp 43633ndash436432018

[31] W Dahech M Patzold and N Youssef ldquoA non-stationarymobile-to-mobile multipath fading channel model takingaccount of velocity variations of the mobile stationsrdquo inProceedings of the IEEE International Symposium on Antennasand Propagation pp 1ndash4 Lisbon Portugal April 2015

[32] W Dahech M Patzold C A Gutierrez and N Youssef ldquoAnon-stationary mobile-to-mobile channel model allowing forvelocity and trajectory variations of the mobile stationsrdquo IEEETransactions on Wireless Communications vol 16 no 3pp 1987ndash2000 2017

[33] C A Gutierrez M Patzold W Dahech and N Youssef ldquoAnon-WSSUS mobile-to-mobile channel model assuming ve-locity variations of the mobile stationsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 1ndash6 San Francisco CA USA March 2017

[34] Q Zhu Y Yang C X Wang et al ldquoSpatial correlations of a3D non-stationary MIMO channel model with 3D antennaarrays and 3D arbitrary trajectoriesrdquo IEEE Wireless Com-munications Letters vol 8 no 2 pp 512ndash515 2019

[35] Y Yang Q Zhu W Li et al ldquoA general 3D non-stationarytwin-cluster model for vehicle-to-vehicle MIMO channelsrdquo inProceedings of the Wireless Communications and SignalProcessing pp 1ndash6 Hangzhou China Octobor 2018

[36] H Hofstertter A F Molisch and N Czink ldquoA twin-clusterMIMO channel modelrdquo in Proceedings of the Antennas andPropagation (EuCAP) pp 1ndash8 Nice France November 2006

[37] I Sen and D W Matolak ldquoVehicle-vehicle channel modelsfor the 5-GHz bandrdquo IEEE Transactions on IntelligentTransportation Systems vol 9 no 2 pp 235ndash245 2008

[38] T Zwick C Fischer D Didascalou and W Wiesbeck ldquoAstochastic spatial channel model based on wave-propagationmodelingrdquo IEEE Journal on Selected Areas in Communica-tions vol 18 no 1 pp 6ndash15 2000

[39] K Mammasis and RW Stewart ldquoe Fisher-Bingham spatialcorrelation model for multielement antenna systemsrdquo IEEETransactions on Vehicular Technology vol 58 no 5pp 2130ndash2136 2009

[40] M Konstantinos S W Robert and J S ompson ldquoSpatialfading correlation model using mixtures of Von Mises Fisherdistributionsrdquo IEEE Transactions on Wireless Communica-tions vol 8 no 4 pp 2046ndash2055 2009

[41] A Fayziyev M Patzold E Masson Y Cocheril andM Berbineau ldquoA measurement-based channel model forvehicular communications in tunnelsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 116ndash121 Istanbul Turkey April 2014

[42] S Payami and F Tufvesson ldquoChannel measurements andanalysis for very large array systems at 26 GHzrdquo in Pro-ceedings of the Antennas and Propagation (EUCAP)pp 433ndash437 Prague Czech Republic March 2012

12 International Journal of Antennas and Propagation

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1113957husn(t) limM⟶infin

1M

1113970

1113944

M

m1ej ΦDnm(t)+ΦLusnm(t)+ΦInm( ) (4)

where M denotes the subpath number ΦInm stands for therandom initial phase distributing over [0 2π) uniformlyand ΦDnm(t) means the phase somehow affected by variantDoppler frequency Here we model ΦDnm(t) as

ΦDnm(t) k 1113946t

0vMTSMT

n tprime( 1113857 middot 1113957sMTnm tprime( 1113857 + vS

MRn MR

tprime( 1113857

middot 1113957sMRnm tprime( 1113857dtprime

(5)

where k 2πfcc denotes the wave number with fc and crepresenting the carrier frequency and light speed re-spectively viSi

n (t) i isin MTMR denotes the relative ve-locity between i and Si

n e departure or arrival angle unit

vector of the mth subpath within the nth path 1113957sinm(t) is

defined as

sinm(t)

cos θinm(t)cos ϕi

nm(t)

cos θinm(t)sin ϕi

nm(t)

sin θinm(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (6)

In (4) ΦLusnm(t) denotes the offset phase caused by themovements of terminals

ΦLusnm(t) k middot 1113957sMTnm(t)1113872 1113873

Tmiddot RMT

v (t) middot dMTt0s + k middot 1113957sMT

nm(t)1113872 1113873T

middot RMTv (t) middot dMTt0

u

(7)where [middot]T means transpose operation and dMTt0

s representssth antenna position at initial time instant Similarly dMRt0

u

θvMR(t)

x~x~

x~

x~vMR(t)

vMR(t)

vMR(t)

vMR(t)

z~ z~

z~

z~

dsMR(t)

dsMR(t)

ϕvMR(t)ϕnm

MR(t)

ϕnmMR(t)

MR(t)θnm

MR(t)θr

MR(t)θnm

y~

y~y~

y~

SnMR

SnMT

vSnMR

(t)

vSnMR

(t)

zz

z

z

y

y

yy

xxx

ϕvMR(t)

Figure 1 Typical 3D V2V communication system

Table 1 Definitions of channel parameters

N(t) Valid path numberPn(t) τn(t) e nth path power and delayM e number of subpaths

dMTs (t) dMR

u (t)Location vectors of transmitter antenna element s and

receiver antenna element u respectively

Li(t) LSin (t)

Location vectors of i and Sin i isin MTMR

respectively

vi(t) vSin (t)

Velocity vectors of i and Sin i isin MTMR

respectively||vi(t)|| ||vS

in (t)|| Speeds of i and Si

n i isin MTMR respectively

ϕiv(t)ϕSi

nv (t)

Traveling azimuth angles of i and Sin i isin MTMR

respectively

θiv(t) θSi

nv (t)

Traveling elevation angles of i and Sin i isin MTMR

respectively

1113957sinm(t)

Spherical unit vector of i along ray m of cluster ni isin MTMR

ϕMTnm(t) θMT

nm(t)Azimuth and elevation departure angles along ray m

of cluster n respectively

ϕMRnm(t) θMR

nm(t)Azimuth and elevation arrival angles along ray m of

cluster n respectively

International Journal of Antennas and Propagation 3

represents the corresponding antenna position at initialcorresponding time Note that Ri

v(t) is a rotation matrix dueto the 3D arbitrary trajectory and it can be written as

Riv(t)

cos θiv(t)cos ϕi

v(t) minus sin ϕiv(t) minus sin θi

v(t)cos ϕiv(t)

cos θiv(t)sin ϕi

v(t) cosϕiv(t) minus sin θi

v(t)sin ϕiv(t)

sin θiv(t) 0 cos θi

v(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)

3 Time-Variant Channel Parameters

31 Path Number e valid path number has been provedto be time-variant under V2V scenarios by the measureddata in [37] due to themoving clusters and two terminals Inother words the old disappearing clusters and newappearing ones are random We model these with a Markovprocess [38] λG is used to denote the birth rate and λR

denotes the death rate e probability of each pathremaining from t to t + Δt is calculated as [19]

Pr(tΔt) eminus λR middotPc vSMT(t)

+ vSMR

(t)

( 1113857Δt

middot e minus λR vMR(t)minus vMT(t) Δt( )

(9)

where Pc means the moving percentage and ||vSi

(t)||i isin MTMR represents clusters average speed Nnew(t) isintroduced to describe newly generated path number andcalculated as

E Nnew(t)1113864 1113865 λG

λR

1 minus Pr(tΔt)( 1113857 (10)

Combining (9) with (10) the averaged path number canbe expressed as

E N(t) N(t minus Δt)Pr(tΔt) + E Nnew(t)1113864 1113865 λG

λR

(11)

32Delays andPowers For the nth valid path the total delayat time instant t consists of the first and last bounce delaysand virtual link delay and it is a function of total distance as

τn(t) LMT(t) minus LS

MTn (t)

+ LMR(t) minus LSMRn (t)

c+ 1113957τn(t)

(12)

where Li(t) and LSin (t) i isin MTMR mean the in-

stantaneous locations of i and Sin and can be updated from

the values of previous time instant by

Li(t) Li

(t minus Δt) + vi(t minus Δt)Δt

LSin (t) LS

in (t minus Δt) + vS

in (t minus Δt)Δt

(13)

where 1113957τn(t) denotes the equivalent delay of virtual link and itcan be updated by the first-order filtering method in [26] asfollows

1113957τn(t) 1113957τn(t minus Δt)eminus Δtτdec( ) + 1 minus eminus Δtτdec( )1113874 1113875X (14)

where X sim U[LMT(t) minus LMR(t)c τmax] in which τmaxdenotes the maximum delay and τdec denotes the decor-relation speed of time-variant delays Based on the mea-surement-based method we can get the averaged power as

Pnprime(t) eminus τn(t) 1minus rDSrDSσDS( ) times 10minus ξn10 (15)

where ξn rDS and σDS mean the shadowing degree delayfactor and delay spread respectively which are all relatedwith the environments Finally all the normalized power canbe denoted as

Pn(t) Pnprime(t)

1113936N(t)n1 Pnprime(t)

(16)

33 Time-Variant Angles Measurements in [39] revealedthat the azimuth and elevation angles at two terminals arenot independent and cannot be depicted by two independentdistributions such as Gaussian or Laplacian In this paper wetake the 3D von MisesndashFisher (VMF) distribution to de-scribe these two joint angles [40] e probability densityfunction (PDF) of VMF distribution can be expressed as [39]

p(ϕ θ) κ cos θeκ(cos θ cos θ cos(ϕminus ϕ)+sin θ sin θ)

4π sinh(κ)

minus π le ϕle π minusπ2le θle

π2

(17)

where ϕ and θ represent the mean values of azimuth andelevation angles respectively and κ is the shape factor andcan be extracted from the measurement data

As long as κ is determined the time-variant angle ofdeparture (AoD) and angle of arrival (AoA) can be obtained byϕ(t) and θ(t) respectively which can be expressed as follows

θi

n(t) arcsinLS

in

z (t) minus Liz(t)

LSin (t) minus Lit)

⎛⎝ ⎞⎠

ϕi

n(t)

arccosLS

in

x (t) minus Lix(t)

LSin (t) minus Li(t)

1113874 1113875cos θi

n(t)1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

LSin

y (t) minus Liy(t)ge 0

minus arccosLS

in

x (t) minus Lix(t)

LSin (t) minus Li(t)

1113874 1113875cos θi

n(t)1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

LSin

y (t) minus Liy(t)lt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

where i isin MTMR Note that κ almost unchanged duringthe short simulation time period Figure 2 gives an exampleof VMF distributed angles of AoA (or AoD) on the unitsphere with different parameters It also reveals that with the

4 International Journal of Antennas and Propagation

increase in shape factor κ the shape of distribution becomesmore concentrated

4 Time-Variant Statistical Properties

e normalized spatial-temporal correlation function(STCF) between two different subchannels can be defined by[16 19 25]

1113957Ru2 s2n

u1 s1n(tΔtΔd) E 1113957hu1 s1n(t)1113957h

lowastu2 s2 n(t + Δt)1113960 1113961

E 1113957hu1 s1 n(t)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877E 1113957hu2 s2 n(t + Δt)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(19)

where E[middot] represents the expectation function ( )lowast iscomplex conjugate and Δt means time lag Moreover Δd

ΔdMTΔdMR1113966 1113967 denotes the space lag and consists of ΔdMT

and ΔdMR which are the corresponding space lags at twoterminals respectively By substituting our proposed

channel model (4) into (19) the expression of STCF can beobtained as

1113957Ru2 s2 n

u1 s1 n(tΔtΔd) limM⟶infin

1

M

1113970

1113944

M

m1ej ΦDu2 s2 nm(t+Δt)+ΦLu2 s2 nm(t+Δt)+ΦInm1113872 1113873eminus j ΦDu1 s1 nm(t)+ΦLu1 s1 nm(t)+ΦInm1113872 1113873

1113946111394611139461113946

θMTn ϕMT

n θMRn ϕMR

n

p ϕMTn (t + Δt) θMT

n (t + Δt)1113872 1113873

1113969

p ϕMRn (t + Δt) θMR

n (t + Δt)1113872 1113873

1113969

middot

p ϕMRn (t) θMR

n (t)1113872 1113873

1113969

middot

p ϕMTn (t) θMT

n (t)1113872 1113873

1113969

ej ΦDu2 s2 n(t+Δt)minus ΦDu1 s1 n(t)+ΦLu2 s2 n(t+Δt)minus ΦLu1 s1 n(t)1113872 1113873dθMT

n dϕMTn dθMR

n dϕMRn

(20)

41 Time-Variant SCFs e SCFs is reduced from STCFwhen the time lagΔt equals to zeroen it can be expressed as

1113957Ru2s2 n

u1s1 n(tΔd) 1113946111394611139461113946

θMTn ϕMT

n θMRn ϕMR

n p ϕMT

n (t) θMTn (t)1113872 1113873p ϕMR

n (t) θMRn (t)1113872 1113873 middot ej ΦLu2 s2 n(t)minus ΦLu1 s1 n(t)1113872 1113873

dθMTn dϕMT

n dθMRn dϕMR

n

(21)

Since the clusters SMTn and SMR

n are independent we canrewrite (21) as

1113957Ru2 s2 n

u1 s1 n(tΔd) 1113957RMTn tΔdMT

1113872 1113873 middot 1113957RMRn tΔdMR

1113872 1113873 (22)

where 1113957Ri

n(tΔdi) i isin MTMR denotes the normalizedSCF at the MT or MR By substituting (7) into (21) thefollowing equation can be obtained

1113957Ri

n tΔdi1113872 1113873 1113946

π

minus π1113946π

minus πejk 1113957s i

n (t)( 1113857T

middotRiV(t)middotΔdit0

middot pϕinm(t)θi

nm(t) ϕin(t) θi

n(t)1113872 1113873dϕindθi

n

(23)

It is shown in (23) that not only the angle of horizontalplane ϕi

n(t) but also the one of vertical plane θin(t) affects the

result of (23) us traditional 2D models cannot obtainaccurate correlation functions in 3D scattering environ-ments By setting θi

n(t) θi

n(t) + ζ and ϕin(t) ϕi

n(t) + υand substituting them into (23) it yields

1113957Ri

n tΔdi1113872 1113873 1113946

1113946

ζ υ

κ cos θi

n(t) + ζ1113874 1113875AiBi

4π sinh(κ)dζdυ

(24)

where

Ai

eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)cos(υ)+sin θi

n(t)+ζ( 1113857sin θi

n(t)( 1113857 (25)

Bi

ejk1113957s ϕi

n(t)+υθi

n(t)+ζ( 1113857RiV(t)Δdit0

(26)

ndash1ndash05ndash05

05

005

1

Xndash1

0

1ndash1

ndash05

0

05

1

Y

Z

κ = 395

θ = ndash6degϕ = 130degndashndash

κ = 2773

θ = 0degϕ = 10degndashndash

κ = 65

θ = 6degϕ = 240degndashndash

Figure 2 e VMF distributed angles on the unit sphere

International Journal of Antennas and Propagation 5

e measurement data demonstrate that the anglespread is usually narrow under V2V communication sce-narios With this assumption we have cos(ζ) asymp cos(υ) asymp 1sin(ζ) asymp ζ sin(υ) asymp υ κcos(υ) asymp κ(1 minus υ22) and κcos(ζ)

asymp κ(1 minus ζ22) Consequently we can rewrite (25) and (26) as

Ai asymp eκ 1minus ζ22minus 2 cos2 θ

i

n(t)( 1113857(υ2)2( 1113857 (27)

Bi asymp ejk 1113957s ϕ

i

n(t)θi

n(t)( 1113857+Ci(t)ζ+Ei(t)υ( 1113857RiV(t)Δdit0

(28)

where

Ci(t) minus sin θ

i

n(t)cosϕi

n(t) minus sin θi

n(t)sin ϕi

n(t) cos θi

n(t)1113876 1113877

(29)

Di(t) minus sinϕi

n(t) cosϕi

n(t) 01113876 1113877 (30)

By substituting (27)ndash(30) into (24) and using Eulerformula we can obtain the following equation

1113957Ri

n tΔdi1113872 1113873 asymp

ejk1113957s ϕi

n(t)θi

n(t)( 1113857RiV(t)Δdit0+κcos( θ

i

n(t)1113857

κ1minus m2( )(2π)m2Im2minus 1(κ)

middot 1113946Δiϕ

minus Δiϕ

eminus Ei(t)υ2 cos kDi(t)Ri

V(t)Δdit0υ1113872 1113873dυ

middot 1113946Δiθ

minus Δiθ

eminus κζ22cos kCi(t)Ri

V(t)Δdit0ζ1113872 1113873dζ

(31)

where Ei(t) κ cos2(θi

n(t))2 and Δiϕ and Δi

θ are the anglespreads of AAoA and EAoA With the help of integrationformula

1113946c

minus ceminus ax2

cos(bx)dx j

π

radiceminus b24a(erfi(b minus 2jac)2

a

radic) minus erfi(b + 2jac2

a

radic)

2a

radic (32)

e approximate closed-form result of SCF is as follows

1113957Ri

n tΔdi1113872 1113873 asymp

minus Im erfi Fi(t)( 1113857( 1113857Im erfi Gi(t)( 1113857( 1113857

8 sinh(κ)

middot ejk1113957s ϕi

n(t)θi

n(t)( 1113857RiV(t)Δdit0

middot eminus kRiV(t)Δdit0( )

2 Di(t)2κ

radic( )

2+ Ci(t)

2κ

radic( )

2( 1113857+κ

(33)

where erfi(middot) is the imaginary error function

Fi(t)

kDi(t)RiV(t)Δdit0 minus jκcos2 θ

i

n(t)1113874 1113875Δiϕ

2κcos2 θi

n(t)1113874 1113875

1113970 (34)

Gi(t)

kCi(t)RiV(t)Δdit0 minus jκΔi

θ2κ

radic (35)

1113957Ru1 s1 n

u1 s1 n(tΔt) 1113946111394611139461113946

θMTn ϕMT

n θMRn ϕMR

n

p ϕMTn (t + Δt) θMT

n (t + Δt)1113872 1113873

1113969

p ϕMRn (t + Δt) θMR

n (t + Δt)1113872 1113873

1113969

p ϕMRn (t) θMR

n (t)1113872 1113873

1113969

middot

p ϕMTn (t) θMT

n (t)1113872 1113873

1113969

ej ΦDu1 s1 n(t)minus ΦDu1 s1 n(t+Δt)+ΦLu1 s1 n(t)minus ΦLu1 s1 n(t+Δt)1113872 1113873dθMT

n dϕMTn dθMR

n dϕMRn

(36)

1113957Ri

n(tΔt) Bϕi

nθin

p ϕin(t) θi

n(t)1113872 1113873

1113969

p ϕin(t + Δt) θi

n(t + Δt)1113872 1113873

1113969

eminus jk 1113946

t+Δt

tviSi

n tprime( 1113857 middot sinm tprime( 1113857dtprime

dϕindθi

n (37)

6 International Journal of Antennas and Propagation

1113957Ri

n(tΔt) κ

4π sinh(κ)1113946Δθ

minus Δθ1113946Δϕ

minus Δϕ

cos θi

n(t) + ζ1113874 1113875eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)( 1113857cos(υ)+sin θi

n(t)+ζ( 1113857sinθi

n(t)( 11138571113970

middot

cos θi

n(t + Δt) + ζ1113874 1113875eκ cos θi

n(t+Δt)+ζ( 1113857cosθi

n(t+Δt)cos(υ)+sin θi

n(t+Δt)+ζ( 1113857sinθi

n(t+Δt)( 11138571113970

middot eminus jk1113938t+Δt

tviSi

n (t)

cos ϕ

iSin

v (t)minus ϕi

n(t)+υ( 11138571113872 1113873cos θi

n(t)+ζ( 1113857cosθiSi

nv (t)+sin θ

i

n(t)+ζ( 1113857sinθiSi

nv (t)1113960 1113961dt

dυdζ

(38)

1113957Ri

n(tΔt) κ

4π sinh(κ)1113946Δθ

minus Δθ1113946Δϕ

minus Δϕ

cos θi

n(t) + ζ1113874 1113875eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)( 1113857cos(υ)+sin θi

n(t)+ζ( 1113857sinθi

n(t)( 11138571113970

middot

cos θi

n(t + Δt) + ζ1113874 1113875eκ cos θi

n(t+Δt)+ζ( 1113857cosθi

n(t+Δt)cos(υ)+sin θi

n(t+Δt)+ζ( 1113857sinθi

n(t+Δt)( 11138571113970

middot eminus jk viSin (t)

cos Ai(tΔtυζ)( )Di(tΔt)+cos Bi(tΔtυζ)( )Ei(tΔt)+sin Ci(tΔtζ)( )Fi(tΔt)[ ]dυdζ

(39)

Finally by substituting (33)ndash(35) into (22) the finalresult of SCF in our proposed model can be obtained Due totime-variant communication environments SCF is alsotime-dependent

42 Time-Variant TCFs e TCF is reduced from STCFwhen Δd is set to be zero en TCF can be derived as(36) It also equals to the product of TCFs at the MT andMR 1113957R

i

n(tΔt) i isin MTMR when the clusters SMTn and

SMRn are independent For simplicity we assumed that theantenna array is placed at the origin of the coordinate

system By substituting (5) and (7) into (36) it can beobtained as (37)

By setting θin(t) θ

i

n(t) + ζ and ϕin(t) ϕi

n(t) + υ andsubstituting them into (37) it yields (38) where ϕiSi

nv (t) and

θiSin

v (t) mean the azimuth and elevation angles of the relativevelocity between i and cluster Si

n respectively It is rea-sonable to assume that the elevation AoDs and azimuthAoDs change linearly during the short time interval asϕi

n(t) k1t + b1 θi

n(t) k2t + b2 where k1 (ϕi

n(t + Δtmax) minus ϕi

n(t))Δtmax k2 (θi

n(t + Δtmax) minus θi

n(t))Δtmaxb1 ϕi

n(t) minus k1t and b2 θi

n(t) minus k2t By integrating theDoppler frequency we can rewrite (38) as (39) where

Ai(tΔt υ ζ)

t + Δt2

1113874 1113875 minus k1 + k2( 1113857 + ϕiSin

v (t) minus b1 + b2 minus υ + ζ

Bi(tΔt υ ζ)

t + Δt2

1113874 1113875 minus k1 minus k2( 1113857 + ϕiSin

v (t) minus b1 minus b2 minus υ minus ζ

Ci(tΔt ζ) k2

t + Δt2

1113874 1113875 + b2 + ζ

Di(tΔt) cos θiSi

n

v (t)sinΔt minus k1 + k2( 11138572( 1113857

minus k1 + k2( 11138571113888 1113889

Ei(tΔt) cos θiSi

n

v (t)sinΔt k1 + k2( 11138572( 1113857

k1 + k2( 11138571113888 1113889

Fi(tΔt) 2sin θiSi

n

v (t)sink2Δt2( 1113857

k21113888 1113889

(40)

1113957Ri

n(tΔt) asympκeκ

4π sinh(κ)

cos θi

n(t)1113874 1113875cos θi

n(t + Δt)1113874 1113875

1113970

eminus jkGi(tΔt)

middot 1113946Δθ

minus Δθeminus κζ22cos kI

i(tΔt)ζ1113872 1113873dζ middot 1113946

Δϕ

minus Δϕeminus Ji(tΔt)υ2 cos kH

i(tΔt)υ1113872 1113873dυ

(41)

International Journal of Antennas and Propagation 7

Measurement data have revealed that small angle spreadsexist under some scenarios In other words κ is usually bigor ζ and υ are small Holding this condition we have

cos(υ minus ζ) asymp 1 cos(υ + ζ) asymp 1 sin(υ minus ζ) asymp υ minus ζ sin(υ + ζ)

asymp υ + ζ cos(ζ) asymp 1 cos(υ) asymp 1 sin(ζ) asymp ζ and sin(υ) asymp υus we can approximate (39) as (41) where

Gi(tΔt) viSi

n (t)

middot Di(tΔt)cos A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)cos B

i(tΔt)1113872 1113873 + viSi

n (t)

middot Fi(tΔt)sin C

i(tΔt)1113872 1113873

Hi(tΔt) viSi

n (t)

middot Di(tΔt)sin A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)sin B

i(tΔt)1113872 1113873

Ii(tΔt) minus viSi

n (t)

middot Di(tΔt)sin A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)sin B

i(tΔt)1113872 1113873 + viSi

n (t)

middot Fi(tΔt)cos C

i(tΔt)1113872 1113873

Ji(tΔt)

κ cos2 θi

n(t)1113874 1113875 + cos2 θi

n(t + Δt)1113874 11138751113874 1113875

4

(42)

1113957Ri

n(tΔt) minusκ

8 sinh(κ)

cos θi

n(t)1113874 1113875cos θi

n(t + Δt)1113874 1113875

1113970

eκminus k2Ii2(tΔt)2κ( 1113857minus k2Hi2(tΔt)4Ji(tΔt)( 1113857minus jkGi(tΔt)

middotIm erfi kHi(tΔt) minus 2jJi(tΔt)Δϕ2

Ji(tΔt)

11139681113872 11138731113872 1113873

Ji(tΔt)

1113968Im erfi kIi(tΔt) minus jκΔθ

2κ

radic( 1113857( 1113857( 1113857

2κ

radic

(43)

where Ai(tΔt) Bi(tΔt) and Ci(tΔt) are the value ofAi(tΔt υ ζ) Bi(tΔt υ ζ) and Ci(tΔt ζ) when υ ζ 0respectively Using the integration formula in (32) theclosed-form expression of (41) can be expressed as (43) Onthis basis we can obtain the final TCF expression of theproposed model

43 Time-Variant DPSDs e DPSD is the Fourier trans-form of TCF 1113957Rusn(tΔt) For the nonstationary aspect of ourproposed model the DPSD can be calculated with short-time Fourier transform as

Sn(f t) 1113946infin

minus infin1113957Rusn(tΔt) eminus j2πfΔt](t minus Δt)dΔt (44)

where window function ](t minus Δt) lasting time is shorter thanthe stationary interval (about several milliseconds [10])

5 Simulation Results and Validation

Firstly in order to evaluate the accuracy of our derived SCFand TCF we calculate the maximum difference between theapproximated and numerical integral results overϕi

n isin [minus 180deg 180deg] and θi

n isin [minus 10deg 10deg] Figure 3 shows theresults of maximum absolute error with different κ nor-malized space lags and time lags As we can see that themaximum absolute error of SCF is less than 0025 when κ ismore than 50 and dλ is less than 3 and it increases as κdecreases or dλ increases Meanwhile the absolute error ofTCF is less than 002 when κ is more than 50 and the time lagis less than 005 s and it increases as κ decreases or the timelag increases Overall the maximum error is acceptable andthus the derived expressions can be used to calculate theSCF and TCF efficiently

Secondly in order to verify the generality of the proposedchannel model ree V2V communication scenarios withdifferent trajectories are selected and given in Figure 4 In thefigure theMTtravels straightly with the fixed velocity and theMR travels in three different velocities with the same de-parture and arrival points Path I has fixed speed and traveldirection and Path II allows speed variation with 2D directionmovement Furthermore Path III has 3D variations in bothvertical and horizontal directions For demonstration pur-pose the distribution of clusters is uniform and we adoptVMF distribution to depict AoAs and AoDs and set κ as395 [39] With the referred data in [41] we assume thatthe speeds of cluster are Gaussian-distributed with 1 kmhmean value and 01 variance Furthermore the carrier fre-quency fc 24GHz ϕS

MTnv ϕS

MRnv θS

MTnv and θS

MRnv are all uni-

formly distributed e former pair distributes over 0 and 2πbut the later one only distributes over minus π36 and π36

It should be mentioned that the V2V channel models in[11ndash26 28 30] only considered fixed velocities of two ter-minals like Path I while the models in [31ndash33] allowed for2D curve trajectories like Path II e absolute values of thetheoretical SCFs of different models including the modelconsidering Path I in [30] and the one focusing on Path II in[32] are simulated and compared in Figure 5 e goodagreement of SCFs between our proposed and other modelsindicates the generalization and compatibility of our pro-posed model In particular the models in [30 32] can beviewed as two special applications in straight or curvedtrajectories

irdly in order to verify the correctness of statisticalproperties of our proposed model under 3D trajectories likePath III the theoretical approximated and simulated valuesof SCFs and TCFs are shown in Figure 6 e x-axis ofFigure 6(a) represents normalized space between antennas

8 International Journal of Antennas and Propagation

320

140

005

120

001

1100 80

0015

60

002

40 200

Max

imum

abso

lute

erro

r

Shape factor ()Normalized antenna

spacing (Δdλ)

(a)

00025

005 200150

10050

Shape factor (κ)Time lag (∆t (s))

0

0005

001

0015

002

Max

imum

abso

lute

erro

r

(b)

Figure 3 Maximum absolute errors between the theoretical and approximate (a) SCFs and (b) TCFs

Start

EndEnd

Start

Path I (1D straight)

Path III (3D arbitrary)

Path II (2D curve)

MTMR

50100 0 ndash50 ndash100 ndash1500

50100

Postion y(m)

Postion x(m)

Posti

on z

(m)

0

1

4

3

5

2

Figure 4 ree typical trajectories of the MR

0

02

04

06

08

1

Model in [28]Model in [30]Proposed model

t = 4st = 8s

t = 0s

Normalized antenna space (Δdλ)0 05 1 15 2

Abso

lute

val

ue o

f SCF

(a)

02

04

06

08

1

Model in [30]Proposed model

00

05 1 15 2

t = 4s

t = 8s

t = 0s

Abso

lute

val

ue o

f SCF

Normalized antenna space (Δdλ)

(b)

Figure 5 Comparison of absolute SCFs for (a) Path I and (b) Path II

International Journal of Antennas and Propagation 9

which is similar at two terminals in the communicationsystem By substituting u1 s1 1 u2 s2 2 and n 1into (21) and (33) we can get the theoretical and approx-imated values of SCFs In addition the measured results in[42] are also shown in Figure 6(a) e good agreement ofSCFs shows the correctness of both the theoretical modeland derivations Similarly by substituting u s 1 and n

1 into (36) and (43) we obtain and show the theoretical andapproximated values of TCFs in Figure 6(b) e goodagreements between theoretical simulated and approxi-mated results reveal that our simulation and derivation arecorrect

Finally substituting TCFs into (44) we can get thetheoretical and simulated values of DPSDs for Path IIIwhich are also shown in Figure 7 It clearly shows thatmoving clusters and two terminals have an influence on thedrifting of DPSDs Moreover good agreements between thesimulated and theoretical results indicate that our proposedmodel is correct

6 Conclusion

is paper has proposed a general channel model charac-terized by 3D scattering environments 3D movements and

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)Measurement

0 05 1 15 2

t = 4s

t = 8s

t = 0s

Normalized antenna space (Δdλ)

Abso

lute

val

ue o

f SCF

(a)

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

0 0005 001 0015 002 0025 003 0035 004

Abso

lute

val

ue o

f TCF

Time lag (Δt)

t = 8s

(b)

Figure 6 Comparison of absolute theoretical and simulated (a) SCFs and (b) TCFs at different time instants

86

4ndash5

0

ndash1000

5

2

10

ndash500

15

0 500 01000Frequency f (HZ)

DPS

D

Time t (s)

(a)

0

5

10

15

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

t = 8s

ndash600 ndash400 ndash200 0 200 400 600Frequency f (HZ)

DPS

D

(b)

Figure 7 (a) eoretical 3D DPSD and (b) theoretical and simulated DPSDs at different time instants for Path III

10 International Journal of Antennas and Propagation

3D-shaped antenna arrays of two terminals for V2V com-munication system e upgraded algorithms for time-evolving parameters such as the number of paths pathdelays path powers and angles have been developed andanalyzed in detail Meanwhile the approximated expressionsof statistical properties including SCF TCF and DPSD havebeen derived and verified by simulations Simulated andanalyzed results show that different moving trajectoriessignificantly affect the V2V channel characteristic Mean-while the generalization and compatibility of our proposedmodel also demonstrate that some previous models with 2Dor even 1D movements have special applications e newmodel can be used to develop analyze and test realistic V2Vcommunication systems in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Key ScientificInstrument and Equipment Development Project (Grant no61827801) Aeronautical Science Foundation of China(Grant no 2017ZC52021) and Open Foundation forGraduate Innovation of NUAA (Grant no KFJJ 20180408)

References

[1] C-X Wang X Cheng and D I Laurenson ldquoVehicle-to-vehicle channel modeling andmeasurements recent advancesand future challengesrdquo IEEE Communications Magazinevol 47 no 11 pp 96ndash103 2009

[2] K Guan B Ai M Liso Nicolas et al ldquoOn the influence ofscattering from traffic signs in vehicle-to-x communicationsrdquoIEEE Transactions on Vehicular Technology vol 65 no 8pp 5835ndash5849 2016

[3] X Cheng C-X Wang B Ai and H Aggoune ldquoEnvelopelevel crossing rate and average fade duration of nonisotropicvehicle-to-vehicle ricean fading channelsrdquo IEEE Transactionson Intelligent Transportation Systems vol 15 no 1 pp 62ndash722014

[4] Q Zhu C Xue X Chen and Y Yang ldquoA newMIMO channelmodel incorporating antenna effectsrdquo Progress In Electro-magnetics Research M vol 50 pp 129ndash140 2016

[5] A G Zajic and G L Stubber ldquoSpace-time correlated mobile-to-mobile channels modelling and simulationrdquo IEEETransactions on Vehicular Technology vol 57 no 2pp 715ndash726 2008

[6] W Fan X Carreno Bautista de Lisbona F Sun J O NielsenM B Knudsen and G F Pedersen ldquoEmulating spatialcharacteristics of MIMO channels for OTA testingrdquo IEEETransactions on Antennas and Propagation vol 61 no 8pp 4306ndash4314 2013

[7] J Zhang Y Zhang Y Yu R Xu Q Zheng and P Zhangldquo3-D MIMO how much does it meet our expectationsobserved from channel measurementsrdquo IEEE Journal on

Selected Areas in Communications vol 35 no 8pp 1887ndash1903 2017

[8] J Karedal F Tufvesson N Czink et al ldquoA geometry-basedstochastic MIMO model for vehicle-to-vehicle communica-tionsrdquo IEEE Transactions onWireless Communications vol 8no 7 pp 3646ndash3657 2009

[9] A Paier T Zemen L Bernado et al ldquoNon-WSSUS vehicularchannel characterization in high-way and urban scenarios at52 GHz using the local scattering functionrdquo in Proceedings ofthe 2008 International ITG Workshop on Smart Antennaspp 9ndash15 Darmstadt Germany February 2008

[10] R He O Renaudin V-M Kolmonen et al ldquoCharacterizationof quasi-stationarity regions for vehicle-to-vehicle radiochannelsrdquo IEEE Transactions on Antennas and Propagationvol 63 no 5 pp 2237ndash2251 2015

[11] YMa L Yang andX Zheng ldquoA geometry-based non-stationaryMIMO channel model for vehicular communicationsrdquo ChinaCommunications vol 15 no 7 pp 30ndash38 2018

[12] C A Gutirrez J T Gutirrez-Mena J M Luna-RiveraD U Campos-Delgado R Velzquez and M Patzold ldquoGe-ometry-based statistical modeling of non-WSSUS mobile-to-mobile Rayleigh fading channelsrdquo IEEE Transactions onVehicular Technology vol 67 no 1 pp 362ndash377 2018

[13] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the Vehicular Technology Confer-ence pp 1ndash6 San Francisco CA USA September 2011

[14] A Borhani and M Patzold ldquoCorrelation and spectralproperties of vehicle-to-vehicle channels in the presence ofmoving scatterersrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 9 pp 4228ndash4239 2013

[15] H Jiang Z Zhang L Wu and J Dang ldquoNovel 3-D irregular-shaped geometry-based channel modeling for semi-ellipsoidvehicle-to-vehicle scattering environmentsrdquo IEEE WirelessCommunications Letters vol 7 no 5 pp 836ndash839 2018

[16] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014

[17] X Zhao X Liang S Li and B Ai ldquoTwo-cylinder and multi-ring GBSSM for realizing and modeling of vehicle-to-vehiclewideband MIMO channelsrdquo IEEE Transactions on IntelligentTransportation Systems vol 17 no 10 pp 2787ndash2799 2016

[18] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA novel 3Dnonstationary channel model based on the von Mises-Fisherscattering distributionrdquo Mobile Information Systemsvol 2016 Article ID 2161460 pp 1ndash9 2016

[19] Y Yuan C-X Wang Y He M M Alwakeel andE-H M Aggoune ldquo3D wideband non-stationary geometry-based stochastic models for non-isotropic MIMO vehicle-to-vehicle channelsrdquo IEEE Transactions on Wireless Communi-cations vol 14 no 12 pp 6883ndash6895 2015

[20] N Avazov S M R Islam D Park and K S Kwak ldquoStatisticalcharacterization of a 3-D propagation model for V2Vchannels in rectangular tunnelsrdquo IEEE Antennas andWirelessPropagation Letters vol 16 pp 2392ndash2395 2017

[21] H Jiang Z Zhang J Dang and L Wu ldquoA novel 3-D massiveMIMO channel model for vehicle-to-vehicle communicationenvironmentsrdquo IEEE Transactions on Communicationsvol 66 no 1 pp 79ndash90 2018

[22] N Avazov S M R Islam D Park and K S Kwak ldquoCluster-based non-stationary channel modeling for vehicle-to-vehiclecommunicationsrdquo IEEE Antennas and Wireless PropagationLetters vol 16 pp 1419ndash1422 2016

International Journal of Antennas and Propagation 11

[23] D Du X Zeng X Jian L Miao and H Wang ldquoree-di-mensional vehicle-to-vehicle channel modeling with multiplemoving scatterersrdquo Mobile Information Systems vol 2017Article ID 7231417 pp 1ndash14 2017

[24] X LiangW Cao and X Zhao ldquoDoppler power spectra for 3Dvehicle-to-vehicle channels with moving scatterersrdquo IEEEAccess vol 6 pp 42822ndash42828 2018

[25] X Chen Y Fang Y Sun Y Pan and W Xiang ldquoStatisticalcharacterization of novel 3D cluster-based MIMO vehicle-to-vehicle models for urban street scattering environmentsrdquoInternational Journal of Antennas and Propagation vol 2018Article ID 6742346 pp 1ndash11 2018

[26] Q Zhu H Li Y Fu et al ldquoA novel 3D non-stationary wirelessMIMO channel simulator and hardware emulatorrdquo IEEETransactions on Communications vol 66 no 9 pp 3865ndash3878 2018

[27] K Jiang X Chen Q Zhu L Chen D Xu and B Chen ldquoAnovel simulation model for nonstationary rice fading chan-nelsrdquo Wireless Communications and Mobile Computingvol 2018 Article ID 8086073 pp 1ndash9 2018

[28] M Patzold C A Gutierrez and N Youssef ldquoOn the con-sistency of non-stationary multipath fading channels withrespect to the average doppler shift and the doppler spreadrdquoin Proceedings of the IEEE Wireless Communications andNetworking Conference (WCNC) pp 1ndash6 San Francisco CAUSA 2017

[29] Q Zhu W Li C-X Wang et al ldquoTemporal correlations for anon-stationary vehicle-to-vehicle channel model allowingvelocity variationsrdquo IEEE Communications Letters vol 23no 7 pp 1280ndash1284 2019

[30] Q Zhu Y Yang X Chen et al ldquoA novel 3D non-stationaryvehicle-to-vehicle channel model and its spatial-temporalcorrelation propertiesrdquo IEEE Access vol 6 pp 43633ndash436432018

[31] W Dahech M Patzold and N Youssef ldquoA non-stationarymobile-to-mobile multipath fading channel model takingaccount of velocity variations of the mobile stationsrdquo inProceedings of the IEEE International Symposium on Antennasand Propagation pp 1ndash4 Lisbon Portugal April 2015

[32] W Dahech M Patzold C A Gutierrez and N Youssef ldquoAnon-stationary mobile-to-mobile channel model allowing forvelocity and trajectory variations of the mobile stationsrdquo IEEETransactions on Wireless Communications vol 16 no 3pp 1987ndash2000 2017

[33] C A Gutierrez M Patzold W Dahech and N Youssef ldquoAnon-WSSUS mobile-to-mobile channel model assuming ve-locity variations of the mobile stationsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 1ndash6 San Francisco CA USA March 2017

[34] Q Zhu Y Yang C X Wang et al ldquoSpatial correlations of a3D non-stationary MIMO channel model with 3D antennaarrays and 3D arbitrary trajectoriesrdquo IEEE Wireless Com-munications Letters vol 8 no 2 pp 512ndash515 2019

[35] Y Yang Q Zhu W Li et al ldquoA general 3D non-stationarytwin-cluster model for vehicle-to-vehicle MIMO channelsrdquo inProceedings of the Wireless Communications and SignalProcessing pp 1ndash6 Hangzhou China Octobor 2018

[36] H Hofstertter A F Molisch and N Czink ldquoA twin-clusterMIMO channel modelrdquo in Proceedings of the Antennas andPropagation (EuCAP) pp 1ndash8 Nice France November 2006

[37] I Sen and D W Matolak ldquoVehicle-vehicle channel modelsfor the 5-GHz bandrdquo IEEE Transactions on IntelligentTransportation Systems vol 9 no 2 pp 235ndash245 2008

[38] T Zwick C Fischer D Didascalou and W Wiesbeck ldquoAstochastic spatial channel model based on wave-propagationmodelingrdquo IEEE Journal on Selected Areas in Communica-tions vol 18 no 1 pp 6ndash15 2000

[39] K Mammasis and RW Stewart ldquoe Fisher-Bingham spatialcorrelation model for multielement antenna systemsrdquo IEEETransactions on Vehicular Technology vol 58 no 5pp 2130ndash2136 2009

[40] M Konstantinos S W Robert and J S ompson ldquoSpatialfading correlation model using mixtures of Von Mises Fisherdistributionsrdquo IEEE Transactions on Wireless Communica-tions vol 8 no 4 pp 2046ndash2055 2009

[41] A Fayziyev M Patzold E Masson Y Cocheril andM Berbineau ldquoA measurement-based channel model forvehicular communications in tunnelsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 116ndash121 Istanbul Turkey April 2014

[42] S Payami and F Tufvesson ldquoChannel measurements andanalysis for very large array systems at 26 GHzrdquo in Pro-ceedings of the Antennas and Propagation (EUCAP)pp 433ndash437 Prague Czech Republic March 2012

12 International Journal of Antennas and Propagation

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represents the corresponding antenna position at initialcorresponding time Note that Ri

v(t) is a rotation matrix dueto the 3D arbitrary trajectory and it can be written as

Riv(t)

cos θiv(t)cos ϕi

v(t) minus sin ϕiv(t) minus sin θi

v(t)cos ϕiv(t)

cos θiv(t)sin ϕi

v(t) cosϕiv(t) minus sin θi

v(t)sin ϕiv(t)

sin θiv(t) 0 cos θi

v(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)

3 Time-Variant Channel Parameters

31 Path Number e valid path number has been provedto be time-variant under V2V scenarios by the measureddata in [37] due to themoving clusters and two terminals Inother words the old disappearing clusters and newappearing ones are random We model these with a Markovprocess [38] λG is used to denote the birth rate and λR

denotes the death rate e probability of each pathremaining from t to t + Δt is calculated as [19]

Pr(tΔt) eminus λR middotPc vSMT(t)

+ vSMR

(t)

( 1113857Δt

middot e minus λR vMR(t)minus vMT(t) Δt( )

(9)

where Pc means the moving percentage and ||vSi

(t)||i isin MTMR represents clusters average speed Nnew(t) isintroduced to describe newly generated path number andcalculated as

E Nnew(t)1113864 1113865 λG

λR

1 minus Pr(tΔt)( 1113857 (10)

Combining (9) with (10) the averaged path number canbe expressed as

E N(t) N(t minus Δt)Pr(tΔt) + E Nnew(t)1113864 1113865 λG

λR

(11)

32Delays andPowers For the nth valid path the total delayat time instant t consists of the first and last bounce delaysand virtual link delay and it is a function of total distance as

τn(t) LMT(t) minus LS

MTn (t)

+ LMR(t) minus LSMRn (t)

c+ 1113957τn(t)

(12)

where Li(t) and LSin (t) i isin MTMR mean the in-

stantaneous locations of i and Sin and can be updated from

the values of previous time instant by

Li(t) Li

(t minus Δt) + vi(t minus Δt)Δt

LSin (t) LS

in (t minus Δt) + vS

in (t minus Δt)Δt

(13)

where 1113957τn(t) denotes the equivalent delay of virtual link and itcan be updated by the first-order filtering method in [26] asfollows

1113957τn(t) 1113957τn(t minus Δt)eminus Δtτdec( ) + 1 minus eminus Δtτdec( )1113874 1113875X (14)

where X sim U[LMT(t) minus LMR(t)c τmax] in which τmaxdenotes the maximum delay and τdec denotes the decor-relation speed of time-variant delays Based on the mea-surement-based method we can get the averaged power as

Pnprime(t) eminus τn(t) 1minus rDSrDSσDS( ) times 10minus ξn10 (15)

where ξn rDS and σDS mean the shadowing degree delayfactor and delay spread respectively which are all relatedwith the environments Finally all the normalized power canbe denoted as

Pn(t) Pnprime(t)

1113936N(t)n1 Pnprime(t)

(16)

33 Time-Variant Angles Measurements in [39] revealedthat the azimuth and elevation angles at two terminals arenot independent and cannot be depicted by two independentdistributions such as Gaussian or Laplacian In this paper wetake the 3D von MisesndashFisher (VMF) distribution to de-scribe these two joint angles [40] e probability densityfunction (PDF) of VMF distribution can be expressed as [39]

p(ϕ θ) κ cos θeκ(cos θ cos θ cos(ϕminus ϕ)+sin θ sin θ)

4π sinh(κ)

minus π le ϕle π minusπ2le θle

π2

(17)

where ϕ and θ represent the mean values of azimuth andelevation angles respectively and κ is the shape factor andcan be extracted from the measurement data

As long as κ is determined the time-variant angle ofdeparture (AoD) and angle of arrival (AoA) can be obtained byϕ(t) and θ(t) respectively which can be expressed as follows

θi

n(t) arcsinLS

in

z (t) minus Liz(t)

LSin (t) minus Lit)

⎛⎝ ⎞⎠

ϕi

n(t)

arccosLS

in

x (t) minus Lix(t)

LSin (t) minus Li(t)

1113874 1113875cos θi

n(t)1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

LSin

y (t) minus Liy(t)ge 0

minus arccosLS

in

x (t) minus Lix(t)

LSin (t) minus Li(t)

1113874 1113875cos θi

n(t)1113874 1113875

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

LSin

y (t) minus Liy(t)lt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

where i isin MTMR Note that κ almost unchanged duringthe short simulation time period Figure 2 gives an exampleof VMF distributed angles of AoA (or AoD) on the unitsphere with different parameters It also reveals that with the

4 International Journal of Antennas and Propagation

increase in shape factor κ the shape of distribution becomesmore concentrated

4 Time-Variant Statistical Properties

e normalized spatial-temporal correlation function(STCF) between two different subchannels can be defined by[16 19 25]

1113957Ru2 s2n

u1 s1n(tΔtΔd) E 1113957hu1 s1n(t)1113957h

lowastu2 s2 n(t + Δt)1113960 1113961

E 1113957hu1 s1 n(t)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877E 1113957hu2 s2 n(t + Δt)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(19)

where E[middot] represents the expectation function ( )lowast iscomplex conjugate and Δt means time lag Moreover Δd

ΔdMTΔdMR1113966 1113967 denotes the space lag and consists of ΔdMT

and ΔdMR which are the corresponding space lags at twoterminals respectively By substituting our proposed

channel model (4) into (19) the expression of STCF can beobtained as

1113957Ru2 s2 n

u1 s1 n(tΔtΔd) limM⟶infin

1

M

1113970

1113944

M

m1ej ΦDu2 s2 nm(t+Δt)+ΦLu2 s2 nm(t+Δt)+ΦInm1113872 1113873eminus j ΦDu1 s1 nm(t)+ΦLu1 s1 nm(t)+ΦInm1113872 1113873

1113946111394611139461113946

θMTn ϕMT

n θMRn ϕMR

n

p ϕMTn (t + Δt) θMT

n (t + Δt)1113872 1113873

1113969

p ϕMRn (t + Δt) θMR

n (t + Δt)1113872 1113873

1113969

middot

p ϕMRn (t) θMR

n (t)1113872 1113873

1113969

middot

p ϕMTn (t) θMT

n (t)1113872 1113873

1113969

ej ΦDu2 s2 n(t+Δt)minus ΦDu1 s1 n(t)+ΦLu2 s2 n(t+Δt)minus ΦLu1 s1 n(t)1113872 1113873dθMT

n dϕMTn dθMR

n dϕMRn

(20)

41 Time-Variant SCFs e SCFs is reduced from STCFwhen the time lagΔt equals to zeroen it can be expressed as

1113957Ru2s2 n

u1s1 n(tΔd) 1113946111394611139461113946

θMTn ϕMT

n θMRn ϕMR

n p ϕMT

n (t) θMTn (t)1113872 1113873p ϕMR

n (t) θMRn (t)1113872 1113873 middot ej ΦLu2 s2 n(t)minus ΦLu1 s1 n(t)1113872 1113873

dθMTn dϕMT

n dθMRn dϕMR

n

(21)

Since the clusters SMTn and SMR

n are independent we canrewrite (21) as

1113957Ru2 s2 n

u1 s1 n(tΔd) 1113957RMTn tΔdMT

1113872 1113873 middot 1113957RMRn tΔdMR

1113872 1113873 (22)

where 1113957Ri

n(tΔdi) i isin MTMR denotes the normalizedSCF at the MT or MR By substituting (7) into (21) thefollowing equation can be obtained

1113957Ri

n tΔdi1113872 1113873 1113946

π

minus π1113946π

minus πejk 1113957s i

n (t)( 1113857T

middotRiV(t)middotΔdit0

middot pϕinm(t)θi

nm(t) ϕin(t) θi

n(t)1113872 1113873dϕindθi

n

(23)

It is shown in (23) that not only the angle of horizontalplane ϕi

n(t) but also the one of vertical plane θin(t) affects the

result of (23) us traditional 2D models cannot obtainaccurate correlation functions in 3D scattering environ-ments By setting θi

n(t) θi

n(t) + ζ and ϕin(t) ϕi

n(t) + υand substituting them into (23) it yields

1113957Ri

n tΔdi1113872 1113873 1113946

1113946

ζ υ

κ cos θi

n(t) + ζ1113874 1113875AiBi

4π sinh(κ)dζdυ

(24)

where

Ai

eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)cos(υ)+sin θi

n(t)+ζ( 1113857sin θi

n(t)( 1113857 (25)

Bi

ejk1113957s ϕi

n(t)+υθi

n(t)+ζ( 1113857RiV(t)Δdit0

(26)

ndash1ndash05ndash05

05

005

1

Xndash1

0

1ndash1

ndash05

0

05

1

Y

Z

κ = 395

θ = ndash6degϕ = 130degndashndash

κ = 2773

θ = 0degϕ = 10degndashndash

κ = 65

θ = 6degϕ = 240degndashndash

Figure 2 e VMF distributed angles on the unit sphere

International Journal of Antennas and Propagation 5

e measurement data demonstrate that the anglespread is usually narrow under V2V communication sce-narios With this assumption we have cos(ζ) asymp cos(υ) asymp 1sin(ζ) asymp ζ sin(υ) asymp υ κcos(υ) asymp κ(1 minus υ22) and κcos(ζ)

asymp κ(1 minus ζ22) Consequently we can rewrite (25) and (26) as

Ai asymp eκ 1minus ζ22minus 2 cos2 θ

i

n(t)( 1113857(υ2)2( 1113857 (27)

Bi asymp ejk 1113957s ϕ

i

n(t)θi

n(t)( 1113857+Ci(t)ζ+Ei(t)υ( 1113857RiV(t)Δdit0

(28)

where

Ci(t) minus sin θ

i

n(t)cosϕi

n(t) minus sin θi

n(t)sin ϕi

n(t) cos θi

n(t)1113876 1113877

(29)

Di(t) minus sinϕi

n(t) cosϕi

n(t) 01113876 1113877 (30)

By substituting (27)ndash(30) into (24) and using Eulerformula we can obtain the following equation

1113957Ri

n tΔdi1113872 1113873 asymp

ejk1113957s ϕi

n(t)θi

n(t)( 1113857RiV(t)Δdit0+κcos( θ

i

n(t)1113857

κ1minus m2( )(2π)m2Im2minus 1(κ)

middot 1113946Δiϕ

minus Δiϕ

eminus Ei(t)υ2 cos kDi(t)Ri

V(t)Δdit0υ1113872 1113873dυ

middot 1113946Δiθ

minus Δiθ

eminus κζ22cos kCi(t)Ri

V(t)Δdit0ζ1113872 1113873dζ

(31)

where Ei(t) κ cos2(θi

n(t))2 and Δiϕ and Δi

θ are the anglespreads of AAoA and EAoA With the help of integrationformula

1113946c

minus ceminus ax2

cos(bx)dx j

π

radiceminus b24a(erfi(b minus 2jac)2

a

radic) minus erfi(b + 2jac2

a

radic)

2a

radic (32)

e approximate closed-form result of SCF is as follows

1113957Ri

n tΔdi1113872 1113873 asymp

minus Im erfi Fi(t)( 1113857( 1113857Im erfi Gi(t)( 1113857( 1113857

8 sinh(κ)

middot ejk1113957s ϕi

n(t)θi

n(t)( 1113857RiV(t)Δdit0

middot eminus kRiV(t)Δdit0( )

2 Di(t)2κ

radic( )

2+ Ci(t)

2κ

radic( )

2( 1113857+κ

(33)

where erfi(middot) is the imaginary error function

Fi(t)

kDi(t)RiV(t)Δdit0 minus jκcos2 θ

i

n(t)1113874 1113875Δiϕ

2κcos2 θi

n(t)1113874 1113875

1113970 (34)

Gi(t)

kCi(t)RiV(t)Δdit0 minus jκΔi

θ2κ

radic (35)

1113957Ru1 s1 n

u1 s1 n(tΔt) 1113946111394611139461113946

θMTn ϕMT

n θMRn ϕMR

n

p ϕMTn (t + Δt) θMT

n (t + Δt)1113872 1113873

1113969

p ϕMRn (t + Δt) θMR

n (t + Δt)1113872 1113873

1113969

p ϕMRn (t) θMR

n (t)1113872 1113873

1113969

middot

p ϕMTn (t) θMT

n (t)1113872 1113873

1113969

ej ΦDu1 s1 n(t)minus ΦDu1 s1 n(t+Δt)+ΦLu1 s1 n(t)minus ΦLu1 s1 n(t+Δt)1113872 1113873dθMT

n dϕMTn dθMR

n dϕMRn

(36)

1113957Ri

n(tΔt) Bϕi

nθin

p ϕin(t) θi

n(t)1113872 1113873

1113969

p ϕin(t + Δt) θi

n(t + Δt)1113872 1113873

1113969

eminus jk 1113946

t+Δt

tviSi

n tprime( 1113857 middot sinm tprime( 1113857dtprime

dϕindθi

n (37)

6 International Journal of Antennas and Propagation

1113957Ri

n(tΔt) κ

4π sinh(κ)1113946Δθ

minus Δθ1113946Δϕ

minus Δϕ

cos θi

n(t) + ζ1113874 1113875eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)( 1113857cos(υ)+sin θi

n(t)+ζ( 1113857sinθi

n(t)( 11138571113970

middot

cos θi

n(t + Δt) + ζ1113874 1113875eκ cos θi

n(t+Δt)+ζ( 1113857cosθi

n(t+Δt)cos(υ)+sin θi

n(t+Δt)+ζ( 1113857sinθi

n(t+Δt)( 11138571113970

middot eminus jk1113938t+Δt

tviSi

n (t)

cos ϕ

iSin

v (t)minus ϕi

n(t)+υ( 11138571113872 1113873cos θi

n(t)+ζ( 1113857cosθiSi

nv (t)+sin θ

i

n(t)+ζ( 1113857sinθiSi

nv (t)1113960 1113961dt

dυdζ

(38)

1113957Ri

n(tΔt) κ

4π sinh(κ)1113946Δθ

minus Δθ1113946Δϕ

minus Δϕ

cos θi

n(t) + ζ1113874 1113875eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)( 1113857cos(υ)+sin θi

n(t)+ζ( 1113857sinθi

n(t)( 11138571113970

middot

cos θi

n(t + Δt) + ζ1113874 1113875eκ cos θi

n(t+Δt)+ζ( 1113857cosθi

n(t+Δt)cos(υ)+sin θi

n(t+Δt)+ζ( 1113857sinθi

n(t+Δt)( 11138571113970

middot eminus jk viSin (t)

cos Ai(tΔtυζ)( )Di(tΔt)+cos Bi(tΔtυζ)( )Ei(tΔt)+sin Ci(tΔtζ)( )Fi(tΔt)[ ]dυdζ

(39)

Finally by substituting (33)ndash(35) into (22) the finalresult of SCF in our proposed model can be obtained Due totime-variant communication environments SCF is alsotime-dependent

42 Time-Variant TCFs e TCF is reduced from STCFwhen Δd is set to be zero en TCF can be derived as(36) It also equals to the product of TCFs at the MT andMR 1113957R

i

n(tΔt) i isin MTMR when the clusters SMTn and

SMRn are independent For simplicity we assumed that theantenna array is placed at the origin of the coordinate

system By substituting (5) and (7) into (36) it can beobtained as (37)

By setting θin(t) θ

i

n(t) + ζ and ϕin(t) ϕi

n(t) + υ andsubstituting them into (37) it yields (38) where ϕiSi

nv (t) and

θiSin

v (t) mean the azimuth and elevation angles of the relativevelocity between i and cluster Si

n respectively It is rea-sonable to assume that the elevation AoDs and azimuthAoDs change linearly during the short time interval asϕi

n(t) k1t + b1 θi

n(t) k2t + b2 where k1 (ϕi

n(t + Δtmax) minus ϕi

n(t))Δtmax k2 (θi

n(t + Δtmax) minus θi

n(t))Δtmaxb1 ϕi

n(t) minus k1t and b2 θi

n(t) minus k2t By integrating theDoppler frequency we can rewrite (38) as (39) where

Ai(tΔt υ ζ)

t + Δt2

1113874 1113875 minus k1 + k2( 1113857 + ϕiSin

v (t) minus b1 + b2 minus υ + ζ

Bi(tΔt υ ζ)

t + Δt2

1113874 1113875 minus k1 minus k2( 1113857 + ϕiSin

v (t) minus b1 minus b2 minus υ minus ζ

Ci(tΔt ζ) k2

t + Δt2

1113874 1113875 + b2 + ζ

Di(tΔt) cos θiSi

n

v (t)sinΔt minus k1 + k2( 11138572( 1113857

minus k1 + k2( 11138571113888 1113889

Ei(tΔt) cos θiSi

n

v (t)sinΔt k1 + k2( 11138572( 1113857

k1 + k2( 11138571113888 1113889

Fi(tΔt) 2sin θiSi

n

v (t)sink2Δt2( 1113857

k21113888 1113889

(40)

1113957Ri

n(tΔt) asympκeκ

4π sinh(κ)

cos θi

n(t)1113874 1113875cos θi

n(t + Δt)1113874 1113875

1113970

eminus jkGi(tΔt)

middot 1113946Δθ

minus Δθeminus κζ22cos kI

i(tΔt)ζ1113872 1113873dζ middot 1113946

Δϕ

minus Δϕeminus Ji(tΔt)υ2 cos kH

i(tΔt)υ1113872 1113873dυ

(41)

International Journal of Antennas and Propagation 7

Measurement data have revealed that small angle spreadsexist under some scenarios In other words κ is usually bigor ζ and υ are small Holding this condition we have

cos(υ minus ζ) asymp 1 cos(υ + ζ) asymp 1 sin(υ minus ζ) asymp υ minus ζ sin(υ + ζ)

asymp υ + ζ cos(ζ) asymp 1 cos(υ) asymp 1 sin(ζ) asymp ζ and sin(υ) asymp υus we can approximate (39) as (41) where

Gi(tΔt) viSi

n (t)

middot Di(tΔt)cos A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)cos B

i(tΔt)1113872 1113873 + viSi

n (t)

middot Fi(tΔt)sin C

i(tΔt)1113872 1113873

Hi(tΔt) viSi

n (t)

middot Di(tΔt)sin A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)sin B

i(tΔt)1113872 1113873

Ii(tΔt) minus viSi

n (t)

middot Di(tΔt)sin A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)sin B

i(tΔt)1113872 1113873 + viSi

n (t)

middot Fi(tΔt)cos C

i(tΔt)1113872 1113873

Ji(tΔt)

κ cos2 θi

n(t)1113874 1113875 + cos2 θi

n(t + Δt)1113874 11138751113874 1113875

4

(42)

1113957Ri

n(tΔt) minusκ

8 sinh(κ)

cos θi

n(t)1113874 1113875cos θi

n(t + Δt)1113874 1113875

1113970

eκminus k2Ii2(tΔt)2κ( 1113857minus k2Hi2(tΔt)4Ji(tΔt)( 1113857minus jkGi(tΔt)

middotIm erfi kHi(tΔt) minus 2jJi(tΔt)Δϕ2

Ji(tΔt)

11139681113872 11138731113872 1113873

Ji(tΔt)

1113968Im erfi kIi(tΔt) minus jκΔθ

2κ

radic( 1113857( 1113857( 1113857

2κ

radic

(43)

where Ai(tΔt) Bi(tΔt) and Ci(tΔt) are the value ofAi(tΔt υ ζ) Bi(tΔt υ ζ) and Ci(tΔt ζ) when υ ζ 0respectively Using the integration formula in (32) theclosed-form expression of (41) can be expressed as (43) Onthis basis we can obtain the final TCF expression of theproposed model

43 Time-Variant DPSDs e DPSD is the Fourier trans-form of TCF 1113957Rusn(tΔt) For the nonstationary aspect of ourproposed model the DPSD can be calculated with short-time Fourier transform as

Sn(f t) 1113946infin

minus infin1113957Rusn(tΔt) eminus j2πfΔt](t minus Δt)dΔt (44)

where window function ](t minus Δt) lasting time is shorter thanthe stationary interval (about several milliseconds [10])

5 Simulation Results and Validation

Firstly in order to evaluate the accuracy of our derived SCFand TCF we calculate the maximum difference between theapproximated and numerical integral results overϕi

n isin [minus 180deg 180deg] and θi

n isin [minus 10deg 10deg] Figure 3 shows theresults of maximum absolute error with different κ nor-malized space lags and time lags As we can see that themaximum absolute error of SCF is less than 0025 when κ ismore than 50 and dλ is less than 3 and it increases as κdecreases or dλ increases Meanwhile the absolute error ofTCF is less than 002 when κ is more than 50 and the time lagis less than 005 s and it increases as κ decreases or the timelag increases Overall the maximum error is acceptable andthus the derived expressions can be used to calculate theSCF and TCF efficiently

Secondly in order to verify the generality of the proposedchannel model ree V2V communication scenarios withdifferent trajectories are selected and given in Figure 4 In thefigure theMTtravels straightly with the fixed velocity and theMR travels in three different velocities with the same de-parture and arrival points Path I has fixed speed and traveldirection and Path II allows speed variation with 2D directionmovement Furthermore Path III has 3D variations in bothvertical and horizontal directions For demonstration pur-pose the distribution of clusters is uniform and we adoptVMF distribution to depict AoAs and AoDs and set κ as395 [39] With the referred data in [41] we assume thatthe speeds of cluster are Gaussian-distributed with 1 kmhmean value and 01 variance Furthermore the carrier fre-quency fc 24GHz ϕS

MTnv ϕS

MRnv θS

MTnv and θS

MRnv are all uni-

formly distributed e former pair distributes over 0 and 2πbut the later one only distributes over minus π36 and π36

It should be mentioned that the V2V channel models in[11ndash26 28 30] only considered fixed velocities of two ter-minals like Path I while the models in [31ndash33] allowed for2D curve trajectories like Path II e absolute values of thetheoretical SCFs of different models including the modelconsidering Path I in [30] and the one focusing on Path II in[32] are simulated and compared in Figure 5 e goodagreement of SCFs between our proposed and other modelsindicates the generalization and compatibility of our pro-posed model In particular the models in [30 32] can beviewed as two special applications in straight or curvedtrajectories

irdly in order to verify the correctness of statisticalproperties of our proposed model under 3D trajectories likePath III the theoretical approximated and simulated valuesof SCFs and TCFs are shown in Figure 6 e x-axis ofFigure 6(a) represents normalized space between antennas

8 International Journal of Antennas and Propagation

320

140

005

120

001

1100 80

0015

60

002

40 200

Max

imum

abso

lute

erro

r

Shape factor ()Normalized antenna

spacing (Δdλ)

(a)

00025

005 200150

10050

Shape factor (κ)Time lag (∆t (s))

0

0005

001

0015

002

Max

imum

abso

lute

erro

r

(b)

Figure 3 Maximum absolute errors between the theoretical and approximate (a) SCFs and (b) TCFs

Start

EndEnd

Start

Path I (1D straight)

Path III (3D arbitrary)

Path II (2D curve)

MTMR

50100 0 ndash50 ndash100 ndash1500

50100

Postion y(m)

Postion x(m)

Posti

on z

(m)

0

1

4

3

5

2

Figure 4 ree typical trajectories of the MR

0

02

04

06

08

1

Model in [28]Model in [30]Proposed model

t = 4st = 8s

t = 0s

Normalized antenna space (Δdλ)0 05 1 15 2

Abso

lute

val

ue o

f SCF

(a)

02

04

06

08

1

Model in [30]Proposed model

00

05 1 15 2

t = 4s

t = 8s

t = 0s

Abso

lute

val

ue o

f SCF

Normalized antenna space (Δdλ)

(b)

Figure 5 Comparison of absolute SCFs for (a) Path I and (b) Path II

International Journal of Antennas and Propagation 9

which is similar at two terminals in the communicationsystem By substituting u1 s1 1 u2 s2 2 and n 1into (21) and (33) we can get the theoretical and approx-imated values of SCFs In addition the measured results in[42] are also shown in Figure 6(a) e good agreement ofSCFs shows the correctness of both the theoretical modeland derivations Similarly by substituting u s 1 and n

1 into (36) and (43) we obtain and show the theoretical andapproximated values of TCFs in Figure 6(b) e goodagreements between theoretical simulated and approxi-mated results reveal that our simulation and derivation arecorrect

Finally substituting TCFs into (44) we can get thetheoretical and simulated values of DPSDs for Path IIIwhich are also shown in Figure 7 It clearly shows thatmoving clusters and two terminals have an influence on thedrifting of DPSDs Moreover good agreements between thesimulated and theoretical results indicate that our proposedmodel is correct

6 Conclusion

is paper has proposed a general channel model charac-terized by 3D scattering environments 3D movements and

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)Measurement

0 05 1 15 2

t = 4s

t = 8s

t = 0s

Normalized antenna space (Δdλ)

Abso

lute

val

ue o

f SCF

(a)

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

0 0005 001 0015 002 0025 003 0035 004

Abso

lute

val

ue o

f TCF

Time lag (Δt)

t = 8s

(b)

Figure 6 Comparison of absolute theoretical and simulated (a) SCFs and (b) TCFs at different time instants

86

4ndash5

0

ndash1000

5

2

10

ndash500

15

0 500 01000Frequency f (HZ)

DPS

D

Time t (s)

(a)

0

5

10

15

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

t = 8s

ndash600 ndash400 ndash200 0 200 400 600Frequency f (HZ)

DPS

D

(b)

Figure 7 (a) eoretical 3D DPSD and (b) theoretical and simulated DPSDs at different time instants for Path III

10 International Journal of Antennas and Propagation

3D-shaped antenna arrays of two terminals for V2V com-munication system e upgraded algorithms for time-evolving parameters such as the number of paths pathdelays path powers and angles have been developed andanalyzed in detail Meanwhile the approximated expressionsof statistical properties including SCF TCF and DPSD havebeen derived and verified by simulations Simulated andanalyzed results show that different moving trajectoriessignificantly affect the V2V channel characteristic Mean-while the generalization and compatibility of our proposedmodel also demonstrate that some previous models with 2Dor even 1D movements have special applications e newmodel can be used to develop analyze and test realistic V2Vcommunication systems in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Key ScientificInstrument and Equipment Development Project (Grant no61827801) Aeronautical Science Foundation of China(Grant no 2017ZC52021) and Open Foundation forGraduate Innovation of NUAA (Grant no KFJJ 20180408)

References

[1] C-X Wang X Cheng and D I Laurenson ldquoVehicle-to-vehicle channel modeling andmeasurements recent advancesand future challengesrdquo IEEE Communications Magazinevol 47 no 11 pp 96ndash103 2009

[2] K Guan B Ai M Liso Nicolas et al ldquoOn the influence ofscattering from traffic signs in vehicle-to-x communicationsrdquoIEEE Transactions on Vehicular Technology vol 65 no 8pp 5835ndash5849 2016

[3] X Cheng C-X Wang B Ai and H Aggoune ldquoEnvelopelevel crossing rate and average fade duration of nonisotropicvehicle-to-vehicle ricean fading channelsrdquo IEEE Transactionson Intelligent Transportation Systems vol 15 no 1 pp 62ndash722014

[4] Q Zhu C Xue X Chen and Y Yang ldquoA newMIMO channelmodel incorporating antenna effectsrdquo Progress In Electro-magnetics Research M vol 50 pp 129ndash140 2016

[5] A G Zajic and G L Stubber ldquoSpace-time correlated mobile-to-mobile channels modelling and simulationrdquo IEEETransactions on Vehicular Technology vol 57 no 2pp 715ndash726 2008

[6] W Fan X Carreno Bautista de Lisbona F Sun J O NielsenM B Knudsen and G F Pedersen ldquoEmulating spatialcharacteristics of MIMO channels for OTA testingrdquo IEEETransactions on Antennas and Propagation vol 61 no 8pp 4306ndash4314 2013

[7] J Zhang Y Zhang Y Yu R Xu Q Zheng and P Zhangldquo3-D MIMO how much does it meet our expectationsobserved from channel measurementsrdquo IEEE Journal on

Selected Areas in Communications vol 35 no 8pp 1887ndash1903 2017

[8] J Karedal F Tufvesson N Czink et al ldquoA geometry-basedstochastic MIMO model for vehicle-to-vehicle communica-tionsrdquo IEEE Transactions onWireless Communications vol 8no 7 pp 3646ndash3657 2009

[9] A Paier T Zemen L Bernado et al ldquoNon-WSSUS vehicularchannel characterization in high-way and urban scenarios at52 GHz using the local scattering functionrdquo in Proceedings ofthe 2008 International ITG Workshop on Smart Antennaspp 9ndash15 Darmstadt Germany February 2008

[10] R He O Renaudin V-M Kolmonen et al ldquoCharacterizationof quasi-stationarity regions for vehicle-to-vehicle radiochannelsrdquo IEEE Transactions on Antennas and Propagationvol 63 no 5 pp 2237ndash2251 2015

[11] YMa L Yang andX Zheng ldquoA geometry-based non-stationaryMIMO channel model for vehicular communicationsrdquo ChinaCommunications vol 15 no 7 pp 30ndash38 2018

[12] C A Gutirrez J T Gutirrez-Mena J M Luna-RiveraD U Campos-Delgado R Velzquez and M Patzold ldquoGe-ometry-based statistical modeling of non-WSSUS mobile-to-mobile Rayleigh fading channelsrdquo IEEE Transactions onVehicular Technology vol 67 no 1 pp 362ndash377 2018

[13] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the Vehicular Technology Confer-ence pp 1ndash6 San Francisco CA USA September 2011

[14] A Borhani and M Patzold ldquoCorrelation and spectralproperties of vehicle-to-vehicle channels in the presence ofmoving scatterersrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 9 pp 4228ndash4239 2013

[15] H Jiang Z Zhang L Wu and J Dang ldquoNovel 3-D irregular-shaped geometry-based channel modeling for semi-ellipsoidvehicle-to-vehicle scattering environmentsrdquo IEEE WirelessCommunications Letters vol 7 no 5 pp 836ndash839 2018

[16] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014

[17] X Zhao X Liang S Li and B Ai ldquoTwo-cylinder and multi-ring GBSSM for realizing and modeling of vehicle-to-vehiclewideband MIMO channelsrdquo IEEE Transactions on IntelligentTransportation Systems vol 17 no 10 pp 2787ndash2799 2016

[18] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA novel 3Dnonstationary channel model based on the von Mises-Fisherscattering distributionrdquo Mobile Information Systemsvol 2016 Article ID 2161460 pp 1ndash9 2016

[19] Y Yuan C-X Wang Y He M M Alwakeel andE-H M Aggoune ldquo3D wideband non-stationary geometry-based stochastic models for non-isotropic MIMO vehicle-to-vehicle channelsrdquo IEEE Transactions on Wireless Communi-cations vol 14 no 12 pp 6883ndash6895 2015

[20] N Avazov S M R Islam D Park and K S Kwak ldquoStatisticalcharacterization of a 3-D propagation model for V2Vchannels in rectangular tunnelsrdquo IEEE Antennas andWirelessPropagation Letters vol 16 pp 2392ndash2395 2017

[21] H Jiang Z Zhang J Dang and L Wu ldquoA novel 3-D massiveMIMO channel model for vehicle-to-vehicle communicationenvironmentsrdquo IEEE Transactions on Communicationsvol 66 no 1 pp 79ndash90 2018

[22] N Avazov S M R Islam D Park and K S Kwak ldquoCluster-based non-stationary channel modeling for vehicle-to-vehiclecommunicationsrdquo IEEE Antennas and Wireless PropagationLetters vol 16 pp 1419ndash1422 2016

International Journal of Antennas and Propagation 11

[23] D Du X Zeng X Jian L Miao and H Wang ldquoree-di-mensional vehicle-to-vehicle channel modeling with multiplemoving scatterersrdquo Mobile Information Systems vol 2017Article ID 7231417 pp 1ndash14 2017

[24] X LiangW Cao and X Zhao ldquoDoppler power spectra for 3Dvehicle-to-vehicle channels with moving scatterersrdquo IEEEAccess vol 6 pp 42822ndash42828 2018

[25] X Chen Y Fang Y Sun Y Pan and W Xiang ldquoStatisticalcharacterization of novel 3D cluster-based MIMO vehicle-to-vehicle models for urban street scattering environmentsrdquoInternational Journal of Antennas and Propagation vol 2018Article ID 6742346 pp 1ndash11 2018

[26] Q Zhu H Li Y Fu et al ldquoA novel 3D non-stationary wirelessMIMO channel simulator and hardware emulatorrdquo IEEETransactions on Communications vol 66 no 9 pp 3865ndash3878 2018

[27] K Jiang X Chen Q Zhu L Chen D Xu and B Chen ldquoAnovel simulation model for nonstationary rice fading chan-nelsrdquo Wireless Communications and Mobile Computingvol 2018 Article ID 8086073 pp 1ndash9 2018

[28] M Patzold C A Gutierrez and N Youssef ldquoOn the con-sistency of non-stationary multipath fading channels withrespect to the average doppler shift and the doppler spreadrdquoin Proceedings of the IEEE Wireless Communications andNetworking Conference (WCNC) pp 1ndash6 San Francisco CAUSA 2017

[29] Q Zhu W Li C-X Wang et al ldquoTemporal correlations for anon-stationary vehicle-to-vehicle channel model allowingvelocity variationsrdquo IEEE Communications Letters vol 23no 7 pp 1280ndash1284 2019

[30] Q Zhu Y Yang X Chen et al ldquoA novel 3D non-stationaryvehicle-to-vehicle channel model and its spatial-temporalcorrelation propertiesrdquo IEEE Access vol 6 pp 43633ndash436432018

[31] W Dahech M Patzold and N Youssef ldquoA non-stationarymobile-to-mobile multipath fading channel model takingaccount of velocity variations of the mobile stationsrdquo inProceedings of the IEEE International Symposium on Antennasand Propagation pp 1ndash4 Lisbon Portugal April 2015

[32] W Dahech M Patzold C A Gutierrez and N Youssef ldquoAnon-stationary mobile-to-mobile channel model allowing forvelocity and trajectory variations of the mobile stationsrdquo IEEETransactions on Wireless Communications vol 16 no 3pp 1987ndash2000 2017

[33] C A Gutierrez M Patzold W Dahech and N Youssef ldquoAnon-WSSUS mobile-to-mobile channel model assuming ve-locity variations of the mobile stationsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 1ndash6 San Francisco CA USA March 2017

[34] Q Zhu Y Yang C X Wang et al ldquoSpatial correlations of a3D non-stationary MIMO channel model with 3D antennaarrays and 3D arbitrary trajectoriesrdquo IEEE Wireless Com-munications Letters vol 8 no 2 pp 512ndash515 2019

[35] Y Yang Q Zhu W Li et al ldquoA general 3D non-stationarytwin-cluster model for vehicle-to-vehicle MIMO channelsrdquo inProceedings of the Wireless Communications and SignalProcessing pp 1ndash6 Hangzhou China Octobor 2018

[36] H Hofstertter A F Molisch and N Czink ldquoA twin-clusterMIMO channel modelrdquo in Proceedings of the Antennas andPropagation (EuCAP) pp 1ndash8 Nice France November 2006

[37] I Sen and D W Matolak ldquoVehicle-vehicle channel modelsfor the 5-GHz bandrdquo IEEE Transactions on IntelligentTransportation Systems vol 9 no 2 pp 235ndash245 2008

[38] T Zwick C Fischer D Didascalou and W Wiesbeck ldquoAstochastic spatial channel model based on wave-propagationmodelingrdquo IEEE Journal on Selected Areas in Communica-tions vol 18 no 1 pp 6ndash15 2000

[39] K Mammasis and RW Stewart ldquoe Fisher-Bingham spatialcorrelation model for multielement antenna systemsrdquo IEEETransactions on Vehicular Technology vol 58 no 5pp 2130ndash2136 2009

[40] M Konstantinos S W Robert and J S ompson ldquoSpatialfading correlation model using mixtures of Von Mises Fisherdistributionsrdquo IEEE Transactions on Wireless Communica-tions vol 8 no 4 pp 2046ndash2055 2009

[41] A Fayziyev M Patzold E Masson Y Cocheril andM Berbineau ldquoA measurement-based channel model forvehicular communications in tunnelsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 116ndash121 Istanbul Turkey April 2014

[42] S Payami and F Tufvesson ldquoChannel measurements andanalysis for very large array systems at 26 GHzrdquo in Pro-ceedings of the Antennas and Propagation (EUCAP)pp 433ndash437 Prague Czech Republic March 2012

12 International Journal of Antennas and Propagation

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increase in shape factor κ the shape of distribution becomesmore concentrated

4 Time-Variant Statistical Properties

e normalized spatial-temporal correlation function(STCF) between two different subchannels can be defined by[16 19 25]

1113957Ru2 s2n

u1 s1n(tΔtΔd) E 1113957hu1 s1n(t)1113957h

lowastu2 s2 n(t + Δt)1113960 1113961

E 1113957hu1 s1 n(t)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877E 1113957hu2 s2 n(t + Δt)11138681113868111386811138681113868

111386811138681113868111386811138682

1113876 1113877

1113970

(19)

where E[middot] represents the expectation function ( )lowast iscomplex conjugate and Δt means time lag Moreover Δd

ΔdMTΔdMR1113966 1113967 denotes the space lag and consists of ΔdMT

and ΔdMR which are the corresponding space lags at twoterminals respectively By substituting our proposed

channel model (4) into (19) the expression of STCF can beobtained as

1113957Ru2 s2 n

u1 s1 n(tΔtΔd) limM⟶infin

1

M

1113970

1113944

M

m1ej ΦDu2 s2 nm(t+Δt)+ΦLu2 s2 nm(t+Δt)+ΦInm1113872 1113873eminus j ΦDu1 s1 nm(t)+ΦLu1 s1 nm(t)+ΦInm1113872 1113873

1113946111394611139461113946

θMTn ϕMT

n θMRn ϕMR

n

p ϕMTn (t + Δt) θMT

n (t + Δt)1113872 1113873

1113969

p ϕMRn (t + Δt) θMR

n (t + Δt)1113872 1113873

1113969

middot

p ϕMRn (t) θMR

n (t)1113872 1113873

1113969

middot

p ϕMTn (t) θMT

n (t)1113872 1113873

1113969

ej ΦDu2 s2 n(t+Δt)minus ΦDu1 s1 n(t)+ΦLu2 s2 n(t+Δt)minus ΦLu1 s1 n(t)1113872 1113873dθMT

n dϕMTn dθMR

n dϕMRn

(20)

41 Time-Variant SCFs e SCFs is reduced from STCFwhen the time lagΔt equals to zeroen it can be expressed as

1113957Ru2s2 n

u1s1 n(tΔd) 1113946111394611139461113946

θMTn ϕMT

n θMRn ϕMR

n p ϕMT

n (t) θMTn (t)1113872 1113873p ϕMR

n (t) θMRn (t)1113872 1113873 middot ej ΦLu2 s2 n(t)minus ΦLu1 s1 n(t)1113872 1113873

dθMTn dϕMT

n dθMRn dϕMR

n

(21)

Since the clusters SMTn and SMR

n are independent we canrewrite (21) as

1113957Ru2 s2 n

u1 s1 n(tΔd) 1113957RMTn tΔdMT

1113872 1113873 middot 1113957RMRn tΔdMR

1113872 1113873 (22)

where 1113957Ri

n(tΔdi) i isin MTMR denotes the normalizedSCF at the MT or MR By substituting (7) into (21) thefollowing equation can be obtained

1113957Ri

n tΔdi1113872 1113873 1113946

π

minus π1113946π

minus πejk 1113957s i

n (t)( 1113857T

middotRiV(t)middotΔdit0

middot pϕinm(t)θi

nm(t) ϕin(t) θi

n(t)1113872 1113873dϕindθi

n

(23)

It is shown in (23) that not only the angle of horizontalplane ϕi

n(t) but also the one of vertical plane θin(t) affects the

result of (23) us traditional 2D models cannot obtainaccurate correlation functions in 3D scattering environ-ments By setting θi

n(t) θi

n(t) + ζ and ϕin(t) ϕi

n(t) + υand substituting them into (23) it yields

1113957Ri

n tΔdi1113872 1113873 1113946

1113946

ζ υ

κ cos θi

n(t) + ζ1113874 1113875AiBi

4π sinh(κ)dζdυ

(24)

where

Ai

eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)cos(υ)+sin θi

n(t)+ζ( 1113857sin θi

n(t)( 1113857 (25)

Bi

ejk1113957s ϕi

n(t)+υθi

n(t)+ζ( 1113857RiV(t)Δdit0

(26)

ndash1ndash05ndash05

05

005

1

Xndash1

0

1ndash1

ndash05

0

05

1

Y

Z

κ = 395

θ = ndash6degϕ = 130degndashndash

κ = 2773

θ = 0degϕ = 10degndashndash

κ = 65

θ = 6degϕ = 240degndashndash

Figure 2 e VMF distributed angles on the unit sphere

International Journal of Antennas and Propagation 5

e measurement data demonstrate that the anglespread is usually narrow under V2V communication sce-narios With this assumption we have cos(ζ) asymp cos(υ) asymp 1sin(ζ) asymp ζ sin(υ) asymp υ κcos(υ) asymp κ(1 minus υ22) and κcos(ζ)

asymp κ(1 minus ζ22) Consequently we can rewrite (25) and (26) as

Ai asymp eκ 1minus ζ22minus 2 cos2 θ

i

n(t)( 1113857(υ2)2( 1113857 (27)

Bi asymp ejk 1113957s ϕ

i

n(t)θi

n(t)( 1113857+Ci(t)ζ+Ei(t)υ( 1113857RiV(t)Δdit0

(28)

where

Ci(t) minus sin θ

i

n(t)cosϕi

n(t) minus sin θi

n(t)sin ϕi

n(t) cos θi

n(t)1113876 1113877

(29)

Di(t) minus sinϕi

n(t) cosϕi

n(t) 01113876 1113877 (30)

By substituting (27)ndash(30) into (24) and using Eulerformula we can obtain the following equation

1113957Ri

n tΔdi1113872 1113873 asymp

ejk1113957s ϕi

n(t)θi

n(t)( 1113857RiV(t)Δdit0+κcos( θ

i

n(t)1113857

κ1minus m2( )(2π)m2Im2minus 1(κ)

middot 1113946Δiϕ

minus Δiϕ

eminus Ei(t)υ2 cos kDi(t)Ri

V(t)Δdit0υ1113872 1113873dυ

middot 1113946Δiθ

minus Δiθ

eminus κζ22cos kCi(t)Ri

V(t)Δdit0ζ1113872 1113873dζ

(31)

where Ei(t) κ cos2(θi

n(t))2 and Δiϕ and Δi

θ are the anglespreads of AAoA and EAoA With the help of integrationformula

1113946c

minus ceminus ax2

cos(bx)dx j

π

radiceminus b24a(erfi(b minus 2jac)2

a

radic) minus erfi(b + 2jac2

a

radic)

2a

radic (32)

e approximate closed-form result of SCF is as follows

1113957Ri

n tΔdi1113872 1113873 asymp

minus Im erfi Fi(t)( 1113857( 1113857Im erfi Gi(t)( 1113857( 1113857

8 sinh(κ)

middot ejk1113957s ϕi

n(t)θi

n(t)( 1113857RiV(t)Δdit0

middot eminus kRiV(t)Δdit0( )

2 Di(t)2κ

radic( )

2+ Ci(t)

2κ

radic( )

2( 1113857+κ

(33)

where erfi(middot) is the imaginary error function

Fi(t)

kDi(t)RiV(t)Δdit0 minus jκcos2 θ

i

n(t)1113874 1113875Δiϕ

2κcos2 θi

n(t)1113874 1113875

1113970 (34)

Gi(t)

kCi(t)RiV(t)Δdit0 minus jκΔi

θ2κ

radic (35)

1113957Ru1 s1 n

u1 s1 n(tΔt) 1113946111394611139461113946

θMTn ϕMT

n θMRn ϕMR

n

p ϕMTn (t + Δt) θMT

n (t + Δt)1113872 1113873

1113969

p ϕMRn (t + Δt) θMR

n (t + Δt)1113872 1113873

1113969

p ϕMRn (t) θMR

n (t)1113872 1113873

1113969

middot

p ϕMTn (t) θMT

n (t)1113872 1113873

1113969

ej ΦDu1 s1 n(t)minus ΦDu1 s1 n(t+Δt)+ΦLu1 s1 n(t)minus ΦLu1 s1 n(t+Δt)1113872 1113873dθMT

n dϕMTn dθMR

n dϕMRn

(36)

1113957Ri

n(tΔt) Bϕi

nθin

p ϕin(t) θi

n(t)1113872 1113873

1113969

p ϕin(t + Δt) θi

n(t + Δt)1113872 1113873

1113969

eminus jk 1113946

t+Δt

tviSi

n tprime( 1113857 middot sinm tprime( 1113857dtprime

dϕindθi

n (37)

6 International Journal of Antennas and Propagation

1113957Ri

n(tΔt) κ

4π sinh(κ)1113946Δθ

minus Δθ1113946Δϕ

minus Δϕ

cos θi

n(t) + ζ1113874 1113875eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)( 1113857cos(υ)+sin θi

n(t)+ζ( 1113857sinθi

n(t)( 11138571113970

middot

cos θi

n(t + Δt) + ζ1113874 1113875eκ cos θi

n(t+Δt)+ζ( 1113857cosθi

n(t+Δt)cos(υ)+sin θi

n(t+Δt)+ζ( 1113857sinθi

n(t+Δt)( 11138571113970

middot eminus jk1113938t+Δt

tviSi

n (t)

cos ϕ

iSin

v (t)minus ϕi

n(t)+υ( 11138571113872 1113873cos θi

n(t)+ζ( 1113857cosθiSi

nv (t)+sin θ

i

n(t)+ζ( 1113857sinθiSi

nv (t)1113960 1113961dt

dυdζ

(38)

1113957Ri

n(tΔt) κ

4π sinh(κ)1113946Δθ

minus Δθ1113946Δϕ

minus Δϕ

cos θi

n(t) + ζ1113874 1113875eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)( 1113857cos(υ)+sin θi

n(t)+ζ( 1113857sinθi

n(t)( 11138571113970

middot

cos θi

n(t + Δt) + ζ1113874 1113875eκ cos θi

n(t+Δt)+ζ( 1113857cosθi

n(t+Δt)cos(υ)+sin θi

n(t+Δt)+ζ( 1113857sinθi

n(t+Δt)( 11138571113970

middot eminus jk viSin (t)

cos Ai(tΔtυζ)( )Di(tΔt)+cos Bi(tΔtυζ)( )Ei(tΔt)+sin Ci(tΔtζ)( )Fi(tΔt)[ ]dυdζ

(39)

Finally by substituting (33)ndash(35) into (22) the finalresult of SCF in our proposed model can be obtained Due totime-variant communication environments SCF is alsotime-dependent

42 Time-Variant TCFs e TCF is reduced from STCFwhen Δd is set to be zero en TCF can be derived as(36) It also equals to the product of TCFs at the MT andMR 1113957R

i

n(tΔt) i isin MTMR when the clusters SMTn and

SMRn are independent For simplicity we assumed that theantenna array is placed at the origin of the coordinate

system By substituting (5) and (7) into (36) it can beobtained as (37)

By setting θin(t) θ

i

n(t) + ζ and ϕin(t) ϕi

n(t) + υ andsubstituting them into (37) it yields (38) where ϕiSi

nv (t) and

θiSin

v (t) mean the azimuth and elevation angles of the relativevelocity between i and cluster Si

n respectively It is rea-sonable to assume that the elevation AoDs and azimuthAoDs change linearly during the short time interval asϕi

n(t) k1t + b1 θi

n(t) k2t + b2 where k1 (ϕi

n(t + Δtmax) minus ϕi

n(t))Δtmax k2 (θi

n(t + Δtmax) minus θi

n(t))Δtmaxb1 ϕi

n(t) minus k1t and b2 θi

n(t) minus k2t By integrating theDoppler frequency we can rewrite (38) as (39) where

Ai(tΔt υ ζ)

t + Δt2

1113874 1113875 minus k1 + k2( 1113857 + ϕiSin

v (t) minus b1 + b2 minus υ + ζ

Bi(tΔt υ ζ)

t + Δt2

1113874 1113875 minus k1 minus k2( 1113857 + ϕiSin

v (t) minus b1 minus b2 minus υ minus ζ

Ci(tΔt ζ) k2

t + Δt2

1113874 1113875 + b2 + ζ

Di(tΔt) cos θiSi

n

v (t)sinΔt minus k1 + k2( 11138572( 1113857

minus k1 + k2( 11138571113888 1113889

Ei(tΔt) cos θiSi

n

v (t)sinΔt k1 + k2( 11138572( 1113857

k1 + k2( 11138571113888 1113889

Fi(tΔt) 2sin θiSi

n

v (t)sink2Δt2( 1113857

k21113888 1113889

(40)

1113957Ri

n(tΔt) asympκeκ

4π sinh(κ)

cos θi

n(t)1113874 1113875cos θi

n(t + Δt)1113874 1113875

1113970

eminus jkGi(tΔt)

middot 1113946Δθ

minus Δθeminus κζ22cos kI

i(tΔt)ζ1113872 1113873dζ middot 1113946

Δϕ

minus Δϕeminus Ji(tΔt)υ2 cos kH

i(tΔt)υ1113872 1113873dυ

(41)

International Journal of Antennas and Propagation 7

Measurement data have revealed that small angle spreadsexist under some scenarios In other words κ is usually bigor ζ and υ are small Holding this condition we have

cos(υ minus ζ) asymp 1 cos(υ + ζ) asymp 1 sin(υ minus ζ) asymp υ minus ζ sin(υ + ζ)

asymp υ + ζ cos(ζ) asymp 1 cos(υ) asymp 1 sin(ζ) asymp ζ and sin(υ) asymp υus we can approximate (39) as (41) where

Gi(tΔt) viSi

n (t)

middot Di(tΔt)cos A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)cos B

i(tΔt)1113872 1113873 + viSi

n (t)

middot Fi(tΔt)sin C

i(tΔt)1113872 1113873

Hi(tΔt) viSi

n (t)

middot Di(tΔt)sin A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)sin B

i(tΔt)1113872 1113873

Ii(tΔt) minus viSi

n (t)

middot Di(tΔt)sin A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)sin B

i(tΔt)1113872 1113873 + viSi

n (t)

middot Fi(tΔt)cos C

i(tΔt)1113872 1113873

Ji(tΔt)

κ cos2 θi

n(t)1113874 1113875 + cos2 θi

n(t + Δt)1113874 11138751113874 1113875

4

(42)

1113957Ri

n(tΔt) minusκ

8 sinh(κ)

cos θi

n(t)1113874 1113875cos θi

n(t + Δt)1113874 1113875

1113970

eκminus k2Ii2(tΔt)2κ( 1113857minus k2Hi2(tΔt)4Ji(tΔt)( 1113857minus jkGi(tΔt)

middotIm erfi kHi(tΔt) minus 2jJi(tΔt)Δϕ2

Ji(tΔt)

11139681113872 11138731113872 1113873

Ji(tΔt)

1113968Im erfi kIi(tΔt) minus jκΔθ

2κ

radic( 1113857( 1113857( 1113857

2κ

radic

(43)

where Ai(tΔt) Bi(tΔt) and Ci(tΔt) are the value ofAi(tΔt υ ζ) Bi(tΔt υ ζ) and Ci(tΔt ζ) when υ ζ 0respectively Using the integration formula in (32) theclosed-form expression of (41) can be expressed as (43) Onthis basis we can obtain the final TCF expression of theproposed model

43 Time-Variant DPSDs e DPSD is the Fourier trans-form of TCF 1113957Rusn(tΔt) For the nonstationary aspect of ourproposed model the DPSD can be calculated with short-time Fourier transform as

Sn(f t) 1113946infin

minus infin1113957Rusn(tΔt) eminus j2πfΔt](t minus Δt)dΔt (44)

where window function ](t minus Δt) lasting time is shorter thanthe stationary interval (about several milliseconds [10])

5 Simulation Results and Validation

Firstly in order to evaluate the accuracy of our derived SCFand TCF we calculate the maximum difference between theapproximated and numerical integral results overϕi

n isin [minus 180deg 180deg] and θi

n isin [minus 10deg 10deg] Figure 3 shows theresults of maximum absolute error with different κ nor-malized space lags and time lags As we can see that themaximum absolute error of SCF is less than 0025 when κ ismore than 50 and dλ is less than 3 and it increases as κdecreases or dλ increases Meanwhile the absolute error ofTCF is less than 002 when κ is more than 50 and the time lagis less than 005 s and it increases as κ decreases or the timelag increases Overall the maximum error is acceptable andthus the derived expressions can be used to calculate theSCF and TCF efficiently

Secondly in order to verify the generality of the proposedchannel model ree V2V communication scenarios withdifferent trajectories are selected and given in Figure 4 In thefigure theMTtravels straightly with the fixed velocity and theMR travels in three different velocities with the same de-parture and arrival points Path I has fixed speed and traveldirection and Path II allows speed variation with 2D directionmovement Furthermore Path III has 3D variations in bothvertical and horizontal directions For demonstration pur-pose the distribution of clusters is uniform and we adoptVMF distribution to depict AoAs and AoDs and set κ as395 [39] With the referred data in [41] we assume thatthe speeds of cluster are Gaussian-distributed with 1 kmhmean value and 01 variance Furthermore the carrier fre-quency fc 24GHz ϕS

MTnv ϕS

MRnv θS

MTnv and θS

MRnv are all uni-

formly distributed e former pair distributes over 0 and 2πbut the later one only distributes over minus π36 and π36

It should be mentioned that the V2V channel models in[11ndash26 28 30] only considered fixed velocities of two ter-minals like Path I while the models in [31ndash33] allowed for2D curve trajectories like Path II e absolute values of thetheoretical SCFs of different models including the modelconsidering Path I in [30] and the one focusing on Path II in[32] are simulated and compared in Figure 5 e goodagreement of SCFs between our proposed and other modelsindicates the generalization and compatibility of our pro-posed model In particular the models in [30 32] can beviewed as two special applications in straight or curvedtrajectories

irdly in order to verify the correctness of statisticalproperties of our proposed model under 3D trajectories likePath III the theoretical approximated and simulated valuesof SCFs and TCFs are shown in Figure 6 e x-axis ofFigure 6(a) represents normalized space between antennas

8 International Journal of Antennas and Propagation

320

140

005

120

001

1100 80

0015

60

002

40 200

Max

imum

abso

lute

erro

r

Shape factor ()Normalized antenna

spacing (Δdλ)

(a)

00025

005 200150

10050

Shape factor (κ)Time lag (∆t (s))

0

0005

001

0015

002

Max

imum

abso

lute

erro

r

(b)

Figure 3 Maximum absolute errors between the theoretical and approximate (a) SCFs and (b) TCFs

Start

EndEnd

Start

Path I (1D straight)

Path III (3D arbitrary)

Path II (2D curve)

MTMR

50100 0 ndash50 ndash100 ndash1500

50100

Postion y(m)

Postion x(m)

Posti

on z

(m)

0

1

4

3

5

2

Figure 4 ree typical trajectories of the MR

0

02

04

06

08

1

Model in [28]Model in [30]Proposed model

t = 4st = 8s

t = 0s

Normalized antenna space (Δdλ)0 05 1 15 2

Abso

lute

val

ue o

f SCF

(a)

02

04

06

08

1

Model in [30]Proposed model

00

05 1 15 2

t = 4s

t = 8s

t = 0s

Abso

lute

val

ue o

f SCF

Normalized antenna space (Δdλ)

(b)

Figure 5 Comparison of absolute SCFs for (a) Path I and (b) Path II

International Journal of Antennas and Propagation 9

which is similar at two terminals in the communicationsystem By substituting u1 s1 1 u2 s2 2 and n 1into (21) and (33) we can get the theoretical and approx-imated values of SCFs In addition the measured results in[42] are also shown in Figure 6(a) e good agreement ofSCFs shows the correctness of both the theoretical modeland derivations Similarly by substituting u s 1 and n

1 into (36) and (43) we obtain and show the theoretical andapproximated values of TCFs in Figure 6(b) e goodagreements between theoretical simulated and approxi-mated results reveal that our simulation and derivation arecorrect

Finally substituting TCFs into (44) we can get thetheoretical and simulated values of DPSDs for Path IIIwhich are also shown in Figure 7 It clearly shows thatmoving clusters and two terminals have an influence on thedrifting of DPSDs Moreover good agreements between thesimulated and theoretical results indicate that our proposedmodel is correct

6 Conclusion

is paper has proposed a general channel model charac-terized by 3D scattering environments 3D movements and

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)Measurement

0 05 1 15 2

t = 4s

t = 8s

t = 0s

Normalized antenna space (Δdλ)

Abso

lute

val

ue o

f SCF

(a)

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

0 0005 001 0015 002 0025 003 0035 004

Abso

lute

val

ue o

f TCF

Time lag (Δt)

t = 8s

(b)

Figure 6 Comparison of absolute theoretical and simulated (a) SCFs and (b) TCFs at different time instants

86

4ndash5

0

ndash1000

5

2

10

ndash500

15

0 500 01000Frequency f (HZ)

DPS

D

Time t (s)

(a)

0

5

10

15

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

t = 8s

ndash600 ndash400 ndash200 0 200 400 600Frequency f (HZ)

DPS

D

(b)

Figure 7 (a) eoretical 3D DPSD and (b) theoretical and simulated DPSDs at different time instants for Path III

10 International Journal of Antennas and Propagation

3D-shaped antenna arrays of two terminals for V2V com-munication system e upgraded algorithms for time-evolving parameters such as the number of paths pathdelays path powers and angles have been developed andanalyzed in detail Meanwhile the approximated expressionsof statistical properties including SCF TCF and DPSD havebeen derived and verified by simulations Simulated andanalyzed results show that different moving trajectoriessignificantly affect the V2V channel characteristic Mean-while the generalization and compatibility of our proposedmodel also demonstrate that some previous models with 2Dor even 1D movements have special applications e newmodel can be used to develop analyze and test realistic V2Vcommunication systems in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Key ScientificInstrument and Equipment Development Project (Grant no61827801) Aeronautical Science Foundation of China(Grant no 2017ZC52021) and Open Foundation forGraduate Innovation of NUAA (Grant no KFJJ 20180408)

References

[1] C-X Wang X Cheng and D I Laurenson ldquoVehicle-to-vehicle channel modeling andmeasurements recent advancesand future challengesrdquo IEEE Communications Magazinevol 47 no 11 pp 96ndash103 2009

[2] K Guan B Ai M Liso Nicolas et al ldquoOn the influence ofscattering from traffic signs in vehicle-to-x communicationsrdquoIEEE Transactions on Vehicular Technology vol 65 no 8pp 5835ndash5849 2016

[3] X Cheng C-X Wang B Ai and H Aggoune ldquoEnvelopelevel crossing rate and average fade duration of nonisotropicvehicle-to-vehicle ricean fading channelsrdquo IEEE Transactionson Intelligent Transportation Systems vol 15 no 1 pp 62ndash722014

[4] Q Zhu C Xue X Chen and Y Yang ldquoA newMIMO channelmodel incorporating antenna effectsrdquo Progress In Electro-magnetics Research M vol 50 pp 129ndash140 2016

[5] A G Zajic and G L Stubber ldquoSpace-time correlated mobile-to-mobile channels modelling and simulationrdquo IEEETransactions on Vehicular Technology vol 57 no 2pp 715ndash726 2008

[6] W Fan X Carreno Bautista de Lisbona F Sun J O NielsenM B Knudsen and G F Pedersen ldquoEmulating spatialcharacteristics of MIMO channels for OTA testingrdquo IEEETransactions on Antennas and Propagation vol 61 no 8pp 4306ndash4314 2013

[7] J Zhang Y Zhang Y Yu R Xu Q Zheng and P Zhangldquo3-D MIMO how much does it meet our expectationsobserved from channel measurementsrdquo IEEE Journal on

Selected Areas in Communications vol 35 no 8pp 1887ndash1903 2017

[8] J Karedal F Tufvesson N Czink et al ldquoA geometry-basedstochastic MIMO model for vehicle-to-vehicle communica-tionsrdquo IEEE Transactions onWireless Communications vol 8no 7 pp 3646ndash3657 2009

[9] A Paier T Zemen L Bernado et al ldquoNon-WSSUS vehicularchannel characterization in high-way and urban scenarios at52 GHz using the local scattering functionrdquo in Proceedings ofthe 2008 International ITG Workshop on Smart Antennaspp 9ndash15 Darmstadt Germany February 2008

[10] R He O Renaudin V-M Kolmonen et al ldquoCharacterizationof quasi-stationarity regions for vehicle-to-vehicle radiochannelsrdquo IEEE Transactions on Antennas and Propagationvol 63 no 5 pp 2237ndash2251 2015

[11] YMa L Yang andX Zheng ldquoA geometry-based non-stationaryMIMO channel model for vehicular communicationsrdquo ChinaCommunications vol 15 no 7 pp 30ndash38 2018

[12] C A Gutirrez J T Gutirrez-Mena J M Luna-RiveraD U Campos-Delgado R Velzquez and M Patzold ldquoGe-ometry-based statistical modeling of non-WSSUS mobile-to-mobile Rayleigh fading channelsrdquo IEEE Transactions onVehicular Technology vol 67 no 1 pp 362ndash377 2018

[13] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the Vehicular Technology Confer-ence pp 1ndash6 San Francisco CA USA September 2011

[14] A Borhani and M Patzold ldquoCorrelation and spectralproperties of vehicle-to-vehicle channels in the presence ofmoving scatterersrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 9 pp 4228ndash4239 2013

[15] H Jiang Z Zhang L Wu and J Dang ldquoNovel 3-D irregular-shaped geometry-based channel modeling for semi-ellipsoidvehicle-to-vehicle scattering environmentsrdquo IEEE WirelessCommunications Letters vol 7 no 5 pp 836ndash839 2018

[16] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014

[17] X Zhao X Liang S Li and B Ai ldquoTwo-cylinder and multi-ring GBSSM for realizing and modeling of vehicle-to-vehiclewideband MIMO channelsrdquo IEEE Transactions on IntelligentTransportation Systems vol 17 no 10 pp 2787ndash2799 2016

[18] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA novel 3Dnonstationary channel model based on the von Mises-Fisherscattering distributionrdquo Mobile Information Systemsvol 2016 Article ID 2161460 pp 1ndash9 2016

[19] Y Yuan C-X Wang Y He M M Alwakeel andE-H M Aggoune ldquo3D wideband non-stationary geometry-based stochastic models for non-isotropic MIMO vehicle-to-vehicle channelsrdquo IEEE Transactions on Wireless Communi-cations vol 14 no 12 pp 6883ndash6895 2015

[20] N Avazov S M R Islam D Park and K S Kwak ldquoStatisticalcharacterization of a 3-D propagation model for V2Vchannels in rectangular tunnelsrdquo IEEE Antennas andWirelessPropagation Letters vol 16 pp 2392ndash2395 2017

[21] H Jiang Z Zhang J Dang and L Wu ldquoA novel 3-D massiveMIMO channel model for vehicle-to-vehicle communicationenvironmentsrdquo IEEE Transactions on Communicationsvol 66 no 1 pp 79ndash90 2018

[22] N Avazov S M R Islam D Park and K S Kwak ldquoCluster-based non-stationary channel modeling for vehicle-to-vehiclecommunicationsrdquo IEEE Antennas and Wireless PropagationLetters vol 16 pp 1419ndash1422 2016

International Journal of Antennas and Propagation 11

[23] D Du X Zeng X Jian L Miao and H Wang ldquoree-di-mensional vehicle-to-vehicle channel modeling with multiplemoving scatterersrdquo Mobile Information Systems vol 2017Article ID 7231417 pp 1ndash14 2017

[24] X LiangW Cao and X Zhao ldquoDoppler power spectra for 3Dvehicle-to-vehicle channels with moving scatterersrdquo IEEEAccess vol 6 pp 42822ndash42828 2018

[25] X Chen Y Fang Y Sun Y Pan and W Xiang ldquoStatisticalcharacterization of novel 3D cluster-based MIMO vehicle-to-vehicle models for urban street scattering environmentsrdquoInternational Journal of Antennas and Propagation vol 2018Article ID 6742346 pp 1ndash11 2018

[26] Q Zhu H Li Y Fu et al ldquoA novel 3D non-stationary wirelessMIMO channel simulator and hardware emulatorrdquo IEEETransactions on Communications vol 66 no 9 pp 3865ndash3878 2018

[27] K Jiang X Chen Q Zhu L Chen D Xu and B Chen ldquoAnovel simulation model for nonstationary rice fading chan-nelsrdquo Wireless Communications and Mobile Computingvol 2018 Article ID 8086073 pp 1ndash9 2018

[28] M Patzold C A Gutierrez and N Youssef ldquoOn the con-sistency of non-stationary multipath fading channels withrespect to the average doppler shift and the doppler spreadrdquoin Proceedings of the IEEE Wireless Communications andNetworking Conference (WCNC) pp 1ndash6 San Francisco CAUSA 2017

[29] Q Zhu W Li C-X Wang et al ldquoTemporal correlations for anon-stationary vehicle-to-vehicle channel model allowingvelocity variationsrdquo IEEE Communications Letters vol 23no 7 pp 1280ndash1284 2019

[30] Q Zhu Y Yang X Chen et al ldquoA novel 3D non-stationaryvehicle-to-vehicle channel model and its spatial-temporalcorrelation propertiesrdquo IEEE Access vol 6 pp 43633ndash436432018

[31] W Dahech M Patzold and N Youssef ldquoA non-stationarymobile-to-mobile multipath fading channel model takingaccount of velocity variations of the mobile stationsrdquo inProceedings of the IEEE International Symposium on Antennasand Propagation pp 1ndash4 Lisbon Portugal April 2015

[32] W Dahech M Patzold C A Gutierrez and N Youssef ldquoAnon-stationary mobile-to-mobile channel model allowing forvelocity and trajectory variations of the mobile stationsrdquo IEEETransactions on Wireless Communications vol 16 no 3pp 1987ndash2000 2017

[33] C A Gutierrez M Patzold W Dahech and N Youssef ldquoAnon-WSSUS mobile-to-mobile channel model assuming ve-locity variations of the mobile stationsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 1ndash6 San Francisco CA USA March 2017

[34] Q Zhu Y Yang C X Wang et al ldquoSpatial correlations of a3D non-stationary MIMO channel model with 3D antennaarrays and 3D arbitrary trajectoriesrdquo IEEE Wireless Com-munications Letters vol 8 no 2 pp 512ndash515 2019

[35] Y Yang Q Zhu W Li et al ldquoA general 3D non-stationarytwin-cluster model for vehicle-to-vehicle MIMO channelsrdquo inProceedings of the Wireless Communications and SignalProcessing pp 1ndash6 Hangzhou China Octobor 2018

[36] H Hofstertter A F Molisch and N Czink ldquoA twin-clusterMIMO channel modelrdquo in Proceedings of the Antennas andPropagation (EuCAP) pp 1ndash8 Nice France November 2006

[37] I Sen and D W Matolak ldquoVehicle-vehicle channel modelsfor the 5-GHz bandrdquo IEEE Transactions on IntelligentTransportation Systems vol 9 no 2 pp 235ndash245 2008

[38] T Zwick C Fischer D Didascalou and W Wiesbeck ldquoAstochastic spatial channel model based on wave-propagationmodelingrdquo IEEE Journal on Selected Areas in Communica-tions vol 18 no 1 pp 6ndash15 2000

[39] K Mammasis and RW Stewart ldquoe Fisher-Bingham spatialcorrelation model for multielement antenna systemsrdquo IEEETransactions on Vehicular Technology vol 58 no 5pp 2130ndash2136 2009

[40] M Konstantinos S W Robert and J S ompson ldquoSpatialfading correlation model using mixtures of Von Mises Fisherdistributionsrdquo IEEE Transactions on Wireless Communica-tions vol 8 no 4 pp 2046ndash2055 2009

[41] A Fayziyev M Patzold E Masson Y Cocheril andM Berbineau ldquoA measurement-based channel model forvehicular communications in tunnelsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 116ndash121 Istanbul Turkey April 2014

[42] S Payami and F Tufvesson ldquoChannel measurements andanalysis for very large array systems at 26 GHzrdquo in Pro-ceedings of the Antennas and Propagation (EUCAP)pp 433ndash437 Prague Czech Republic March 2012

12 International Journal of Antennas and Propagation

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Submit your manuscripts atwwwhindawicom

e measurement data demonstrate that the anglespread is usually narrow under V2V communication sce-narios With this assumption we have cos(ζ) asymp cos(υ) asymp 1sin(ζ) asymp ζ sin(υ) asymp υ κcos(υ) asymp κ(1 minus υ22) and κcos(ζ)

asymp κ(1 minus ζ22) Consequently we can rewrite (25) and (26) as

Ai asymp eκ 1minus ζ22minus 2 cos2 θ

i

n(t)( 1113857(υ2)2( 1113857 (27)

Bi asymp ejk 1113957s ϕ

i

n(t)θi

n(t)( 1113857+Ci(t)ζ+Ei(t)υ( 1113857RiV(t)Δdit0

(28)

where

Ci(t) minus sin θ

i

n(t)cosϕi

n(t) minus sin θi

n(t)sin ϕi

n(t) cos θi

n(t)1113876 1113877

(29)

Di(t) minus sinϕi

n(t) cosϕi

n(t) 01113876 1113877 (30)

By substituting (27)ndash(30) into (24) and using Eulerformula we can obtain the following equation

1113957Ri

n tΔdi1113872 1113873 asymp

ejk1113957s ϕi

n(t)θi

n(t)( 1113857RiV(t)Δdit0+κcos( θ

i

n(t)1113857

κ1minus m2( )(2π)m2Im2minus 1(κ)

middot 1113946Δiϕ

minus Δiϕ

eminus Ei(t)υ2 cos kDi(t)Ri

V(t)Δdit0υ1113872 1113873dυ

middot 1113946Δiθ

minus Δiθ

eminus κζ22cos kCi(t)Ri

V(t)Δdit0ζ1113872 1113873dζ

(31)

where Ei(t) κ cos2(θi

n(t))2 and Δiϕ and Δi

θ are the anglespreads of AAoA and EAoA With the help of integrationformula

1113946c

minus ceminus ax2

cos(bx)dx j

π

radiceminus b24a(erfi(b minus 2jac)2

a

radic) minus erfi(b + 2jac2

a

radic)

2a

radic (32)

e approximate closed-form result of SCF is as follows

1113957Ri

n tΔdi1113872 1113873 asymp

minus Im erfi Fi(t)( 1113857( 1113857Im erfi Gi(t)( 1113857( 1113857

8 sinh(κ)

middot ejk1113957s ϕi

n(t)θi

n(t)( 1113857RiV(t)Δdit0

middot eminus kRiV(t)Δdit0( )

2 Di(t)2κ

radic( )

2+ Ci(t)

2κ

radic( )

2( 1113857+κ

(33)

where erfi(middot) is the imaginary error function

Fi(t)

kDi(t)RiV(t)Δdit0 minus jκcos2 θ

i

n(t)1113874 1113875Δiϕ

2κcos2 θi

n(t)1113874 1113875

1113970 (34)

Gi(t)

kCi(t)RiV(t)Δdit0 minus jκΔi

θ2κ

radic (35)

1113957Ru1 s1 n

u1 s1 n(tΔt) 1113946111394611139461113946

θMTn ϕMT

n θMRn ϕMR

n

p ϕMTn (t + Δt) θMT

n (t + Δt)1113872 1113873

1113969

p ϕMRn (t + Δt) θMR

n (t + Δt)1113872 1113873

1113969

p ϕMRn (t) θMR

n (t)1113872 1113873

1113969

middot

p ϕMTn (t) θMT

n (t)1113872 1113873

1113969

ej ΦDu1 s1 n(t)minus ΦDu1 s1 n(t+Δt)+ΦLu1 s1 n(t)minus ΦLu1 s1 n(t+Δt)1113872 1113873dθMT

n dϕMTn dθMR

n dϕMRn

(36)

1113957Ri

n(tΔt) Bϕi

nθin

p ϕin(t) θi

n(t)1113872 1113873

1113969

p ϕin(t + Δt) θi

n(t + Δt)1113872 1113873

1113969

eminus jk 1113946

t+Δt

tviSi

n tprime( 1113857 middot sinm tprime( 1113857dtprime

dϕindθi

n (37)

6 International Journal of Antennas and Propagation

1113957Ri

n(tΔt) κ

4π sinh(κ)1113946Δθ

minus Δθ1113946Δϕ

minus Δϕ

cos θi

n(t) + ζ1113874 1113875eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)( 1113857cos(υ)+sin θi

n(t)+ζ( 1113857sinθi

n(t)( 11138571113970

middot

cos θi

n(t + Δt) + ζ1113874 1113875eκ cos θi

n(t+Δt)+ζ( 1113857cosθi

n(t+Δt)cos(υ)+sin θi

n(t+Δt)+ζ( 1113857sinθi

n(t+Δt)( 11138571113970

middot eminus jk1113938t+Δt

tviSi

n (t)

cos ϕ

iSin

v (t)minus ϕi

n(t)+υ( 11138571113872 1113873cos θi

n(t)+ζ( 1113857cosθiSi

nv (t)+sin θ

i

n(t)+ζ( 1113857sinθiSi

nv (t)1113960 1113961dt

dυdζ

(38)

1113957Ri

n(tΔt) κ

4π sinh(κ)1113946Δθ

minus Δθ1113946Δϕ

minus Δϕ

cos θi

n(t) + ζ1113874 1113875eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)( 1113857cos(υ)+sin θi

n(t)+ζ( 1113857sinθi

n(t)( 11138571113970

middot

cos θi

n(t + Δt) + ζ1113874 1113875eκ cos θi

n(t+Δt)+ζ( 1113857cosθi

n(t+Δt)cos(υ)+sin θi

n(t+Δt)+ζ( 1113857sinθi

n(t+Δt)( 11138571113970

middot eminus jk viSin (t)

cos Ai(tΔtυζ)( )Di(tΔt)+cos Bi(tΔtυζ)( )Ei(tΔt)+sin Ci(tΔtζ)( )Fi(tΔt)[ ]dυdζ

(39)

Finally by substituting (33)ndash(35) into (22) the finalresult of SCF in our proposed model can be obtained Due totime-variant communication environments SCF is alsotime-dependent

42 Time-Variant TCFs e TCF is reduced from STCFwhen Δd is set to be zero en TCF can be derived as(36) It also equals to the product of TCFs at the MT andMR 1113957R

i

n(tΔt) i isin MTMR when the clusters SMTn and

SMRn are independent For simplicity we assumed that theantenna array is placed at the origin of the coordinate

system By substituting (5) and (7) into (36) it can beobtained as (37)

By setting θin(t) θ

i

n(t) + ζ and ϕin(t) ϕi

n(t) + υ andsubstituting them into (37) it yields (38) where ϕiSi

nv (t) and

θiSin

v (t) mean the azimuth and elevation angles of the relativevelocity between i and cluster Si

n respectively It is rea-sonable to assume that the elevation AoDs and azimuthAoDs change linearly during the short time interval asϕi

n(t) k1t + b1 θi

n(t) k2t + b2 where k1 (ϕi

n(t + Δtmax) minus ϕi

n(t))Δtmax k2 (θi

n(t + Δtmax) minus θi

n(t))Δtmaxb1 ϕi

n(t) minus k1t and b2 θi

n(t) minus k2t By integrating theDoppler frequency we can rewrite (38) as (39) where

Ai(tΔt υ ζ)

t + Δt2

1113874 1113875 minus k1 + k2( 1113857 + ϕiSin

v (t) minus b1 + b2 minus υ + ζ

Bi(tΔt υ ζ)

t + Δt2

1113874 1113875 minus k1 minus k2( 1113857 + ϕiSin

v (t) minus b1 minus b2 minus υ minus ζ

Ci(tΔt ζ) k2

t + Δt2

1113874 1113875 + b2 + ζ

Di(tΔt) cos θiSi

n

v (t)sinΔt minus k1 + k2( 11138572( 1113857

minus k1 + k2( 11138571113888 1113889

Ei(tΔt) cos θiSi

n

v (t)sinΔt k1 + k2( 11138572( 1113857

k1 + k2( 11138571113888 1113889

Fi(tΔt) 2sin θiSi

n

v (t)sink2Δt2( 1113857

k21113888 1113889

(40)

1113957Ri

n(tΔt) asympκeκ

4π sinh(κ)

cos θi

n(t)1113874 1113875cos θi

n(t + Δt)1113874 1113875

1113970

eminus jkGi(tΔt)

middot 1113946Δθ

minus Δθeminus κζ22cos kI

i(tΔt)ζ1113872 1113873dζ middot 1113946

Δϕ

minus Δϕeminus Ji(tΔt)υ2 cos kH

i(tΔt)υ1113872 1113873dυ

(41)

International Journal of Antennas and Propagation 7

Measurement data have revealed that small angle spreadsexist under some scenarios In other words κ is usually bigor ζ and υ are small Holding this condition we have

cos(υ minus ζ) asymp 1 cos(υ + ζ) asymp 1 sin(υ minus ζ) asymp υ minus ζ sin(υ + ζ)

asymp υ + ζ cos(ζ) asymp 1 cos(υ) asymp 1 sin(ζ) asymp ζ and sin(υ) asymp υus we can approximate (39) as (41) where

Gi(tΔt) viSi

n (t)

middot Di(tΔt)cos A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)cos B

i(tΔt)1113872 1113873 + viSi

n (t)

middot Fi(tΔt)sin C

i(tΔt)1113872 1113873

Hi(tΔt) viSi

n (t)

middot Di(tΔt)sin A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)sin B

i(tΔt)1113872 1113873

Ii(tΔt) minus viSi

n (t)

middot Di(tΔt)sin A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)sin B

i(tΔt)1113872 1113873 + viSi

n (t)

middot Fi(tΔt)cos C

i(tΔt)1113872 1113873

Ji(tΔt)

κ cos2 θi

n(t)1113874 1113875 + cos2 θi

n(t + Δt)1113874 11138751113874 1113875

4

(42)

1113957Ri

n(tΔt) minusκ

8 sinh(κ)

cos θi

n(t)1113874 1113875cos θi

n(t + Δt)1113874 1113875

1113970

eκminus k2Ii2(tΔt)2κ( 1113857minus k2Hi2(tΔt)4Ji(tΔt)( 1113857minus jkGi(tΔt)

middotIm erfi kHi(tΔt) minus 2jJi(tΔt)Δϕ2

Ji(tΔt)

11139681113872 11138731113872 1113873

Ji(tΔt)

1113968Im erfi kIi(tΔt) minus jκΔθ

2κ

radic( 1113857( 1113857( 1113857

2κ

radic

(43)

where Ai(tΔt) Bi(tΔt) and Ci(tΔt) are the value ofAi(tΔt υ ζ) Bi(tΔt υ ζ) and Ci(tΔt ζ) when υ ζ 0respectively Using the integration formula in (32) theclosed-form expression of (41) can be expressed as (43) Onthis basis we can obtain the final TCF expression of theproposed model

43 Time-Variant DPSDs e DPSD is the Fourier trans-form of TCF 1113957Rusn(tΔt) For the nonstationary aspect of ourproposed model the DPSD can be calculated with short-time Fourier transform as

Sn(f t) 1113946infin

minus infin1113957Rusn(tΔt) eminus j2πfΔt](t minus Δt)dΔt (44)

where window function ](t minus Δt) lasting time is shorter thanthe stationary interval (about several milliseconds [10])

5 Simulation Results and Validation

Firstly in order to evaluate the accuracy of our derived SCFand TCF we calculate the maximum difference between theapproximated and numerical integral results overϕi

n isin [minus 180deg 180deg] and θi

n isin [minus 10deg 10deg] Figure 3 shows theresults of maximum absolute error with different κ nor-malized space lags and time lags As we can see that themaximum absolute error of SCF is less than 0025 when κ ismore than 50 and dλ is less than 3 and it increases as κdecreases or dλ increases Meanwhile the absolute error ofTCF is less than 002 when κ is more than 50 and the time lagis less than 005 s and it increases as κ decreases or the timelag increases Overall the maximum error is acceptable andthus the derived expressions can be used to calculate theSCF and TCF efficiently

Secondly in order to verify the generality of the proposedchannel model ree V2V communication scenarios withdifferent trajectories are selected and given in Figure 4 In thefigure theMTtravels straightly with the fixed velocity and theMR travels in three different velocities with the same de-parture and arrival points Path I has fixed speed and traveldirection and Path II allows speed variation with 2D directionmovement Furthermore Path III has 3D variations in bothvertical and horizontal directions For demonstration pur-pose the distribution of clusters is uniform and we adoptVMF distribution to depict AoAs and AoDs and set κ as395 [39] With the referred data in [41] we assume thatthe speeds of cluster are Gaussian-distributed with 1 kmhmean value and 01 variance Furthermore the carrier fre-quency fc 24GHz ϕS

MTnv ϕS

MRnv θS

MTnv and θS

MRnv are all uni-

formly distributed e former pair distributes over 0 and 2πbut the later one only distributes over minus π36 and π36

It should be mentioned that the V2V channel models in[11ndash26 28 30] only considered fixed velocities of two ter-minals like Path I while the models in [31ndash33] allowed for2D curve trajectories like Path II e absolute values of thetheoretical SCFs of different models including the modelconsidering Path I in [30] and the one focusing on Path II in[32] are simulated and compared in Figure 5 e goodagreement of SCFs between our proposed and other modelsindicates the generalization and compatibility of our pro-posed model In particular the models in [30 32] can beviewed as two special applications in straight or curvedtrajectories

irdly in order to verify the correctness of statisticalproperties of our proposed model under 3D trajectories likePath III the theoretical approximated and simulated valuesof SCFs and TCFs are shown in Figure 6 e x-axis ofFigure 6(a) represents normalized space between antennas

8 International Journal of Antennas and Propagation

320

140

005

120

001

1100 80

0015

60

002

40 200

Max

imum

abso

lute

erro

r

Shape factor ()Normalized antenna

spacing (Δdλ)

(a)

00025

005 200150

10050

Shape factor (κ)Time lag (∆t (s))

0

0005

001

0015

002

Max

imum

abso

lute

erro

r

(b)

Figure 3 Maximum absolute errors between the theoretical and approximate (a) SCFs and (b) TCFs

Start

EndEnd

Start

Path I (1D straight)

Path III (3D arbitrary)

Path II (2D curve)

MTMR

50100 0 ndash50 ndash100 ndash1500

50100

Postion y(m)

Postion x(m)

Posti

on z

(m)

0

1

4

3

5

2

Figure 4 ree typical trajectories of the MR

0

02

04

06

08

1

Model in [28]Model in [30]Proposed model

t = 4st = 8s

t = 0s

Normalized antenna space (Δdλ)0 05 1 15 2

Abso

lute

val

ue o

f SCF

(a)

02

04

06

08

1

Model in [30]Proposed model

00

05 1 15 2

t = 4s

t = 8s

t = 0s

Abso

lute

val

ue o

f SCF

Normalized antenna space (Δdλ)

(b)

Figure 5 Comparison of absolute SCFs for (a) Path I and (b) Path II

International Journal of Antennas and Propagation 9

which is similar at two terminals in the communicationsystem By substituting u1 s1 1 u2 s2 2 and n 1into (21) and (33) we can get the theoretical and approx-imated values of SCFs In addition the measured results in[42] are also shown in Figure 6(a) e good agreement ofSCFs shows the correctness of both the theoretical modeland derivations Similarly by substituting u s 1 and n

1 into (36) and (43) we obtain and show the theoretical andapproximated values of TCFs in Figure 6(b) e goodagreements between theoretical simulated and approxi-mated results reveal that our simulation and derivation arecorrect

Finally substituting TCFs into (44) we can get thetheoretical and simulated values of DPSDs for Path IIIwhich are also shown in Figure 7 It clearly shows thatmoving clusters and two terminals have an influence on thedrifting of DPSDs Moreover good agreements between thesimulated and theoretical results indicate that our proposedmodel is correct

6 Conclusion

is paper has proposed a general channel model charac-terized by 3D scattering environments 3D movements and

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)Measurement

0 05 1 15 2

t = 4s

t = 8s

t = 0s

Normalized antenna space (Δdλ)

Abso

lute

val

ue o

f SCF

(a)

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

0 0005 001 0015 002 0025 003 0035 004

Abso

lute

val

ue o

f TCF

Time lag (Δt)

t = 8s

(b)

Figure 6 Comparison of absolute theoretical and simulated (a) SCFs and (b) TCFs at different time instants

86

4ndash5

0

ndash1000

5

2

10

ndash500

15

0 500 01000Frequency f (HZ)

DPS

D

Time t (s)

(a)

0

5

10

15

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

t = 8s

ndash600 ndash400 ndash200 0 200 400 600Frequency f (HZ)

DPS

D

(b)

Figure 7 (a) eoretical 3D DPSD and (b) theoretical and simulated DPSDs at different time instants for Path III

10 International Journal of Antennas and Propagation

3D-shaped antenna arrays of two terminals for V2V com-munication system e upgraded algorithms for time-evolving parameters such as the number of paths pathdelays path powers and angles have been developed andanalyzed in detail Meanwhile the approximated expressionsof statistical properties including SCF TCF and DPSD havebeen derived and verified by simulations Simulated andanalyzed results show that different moving trajectoriessignificantly affect the V2V channel characteristic Mean-while the generalization and compatibility of our proposedmodel also demonstrate that some previous models with 2Dor even 1D movements have special applications e newmodel can be used to develop analyze and test realistic V2Vcommunication systems in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Key ScientificInstrument and Equipment Development Project (Grant no61827801) Aeronautical Science Foundation of China(Grant no 2017ZC52021) and Open Foundation forGraduate Innovation of NUAA (Grant no KFJJ 20180408)

References

[1] C-X Wang X Cheng and D I Laurenson ldquoVehicle-to-vehicle channel modeling andmeasurements recent advancesand future challengesrdquo IEEE Communications Magazinevol 47 no 11 pp 96ndash103 2009

[2] K Guan B Ai M Liso Nicolas et al ldquoOn the influence ofscattering from traffic signs in vehicle-to-x communicationsrdquoIEEE Transactions on Vehicular Technology vol 65 no 8pp 5835ndash5849 2016

[3] X Cheng C-X Wang B Ai and H Aggoune ldquoEnvelopelevel crossing rate and average fade duration of nonisotropicvehicle-to-vehicle ricean fading channelsrdquo IEEE Transactionson Intelligent Transportation Systems vol 15 no 1 pp 62ndash722014

[4] Q Zhu C Xue X Chen and Y Yang ldquoA newMIMO channelmodel incorporating antenna effectsrdquo Progress In Electro-magnetics Research M vol 50 pp 129ndash140 2016

[5] A G Zajic and G L Stubber ldquoSpace-time correlated mobile-to-mobile channels modelling and simulationrdquo IEEETransactions on Vehicular Technology vol 57 no 2pp 715ndash726 2008

[6] W Fan X Carreno Bautista de Lisbona F Sun J O NielsenM B Knudsen and G F Pedersen ldquoEmulating spatialcharacteristics of MIMO channels for OTA testingrdquo IEEETransactions on Antennas and Propagation vol 61 no 8pp 4306ndash4314 2013

[7] J Zhang Y Zhang Y Yu R Xu Q Zheng and P Zhangldquo3-D MIMO how much does it meet our expectationsobserved from channel measurementsrdquo IEEE Journal on

Selected Areas in Communications vol 35 no 8pp 1887ndash1903 2017

[8] J Karedal F Tufvesson N Czink et al ldquoA geometry-basedstochastic MIMO model for vehicle-to-vehicle communica-tionsrdquo IEEE Transactions onWireless Communications vol 8no 7 pp 3646ndash3657 2009

[9] A Paier T Zemen L Bernado et al ldquoNon-WSSUS vehicularchannel characterization in high-way and urban scenarios at52 GHz using the local scattering functionrdquo in Proceedings ofthe 2008 International ITG Workshop on Smart Antennaspp 9ndash15 Darmstadt Germany February 2008

[10] R He O Renaudin V-M Kolmonen et al ldquoCharacterizationof quasi-stationarity regions for vehicle-to-vehicle radiochannelsrdquo IEEE Transactions on Antennas and Propagationvol 63 no 5 pp 2237ndash2251 2015

[11] YMa L Yang andX Zheng ldquoA geometry-based non-stationaryMIMO channel model for vehicular communicationsrdquo ChinaCommunications vol 15 no 7 pp 30ndash38 2018

[12] C A Gutirrez J T Gutirrez-Mena J M Luna-RiveraD U Campos-Delgado R Velzquez and M Patzold ldquoGe-ometry-based statistical modeling of non-WSSUS mobile-to-mobile Rayleigh fading channelsrdquo IEEE Transactions onVehicular Technology vol 67 no 1 pp 362ndash377 2018

[13] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the Vehicular Technology Confer-ence pp 1ndash6 San Francisco CA USA September 2011

[14] A Borhani and M Patzold ldquoCorrelation and spectralproperties of vehicle-to-vehicle channels in the presence ofmoving scatterersrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 9 pp 4228ndash4239 2013

[15] H Jiang Z Zhang L Wu and J Dang ldquoNovel 3-D irregular-shaped geometry-based channel modeling for semi-ellipsoidvehicle-to-vehicle scattering environmentsrdquo IEEE WirelessCommunications Letters vol 7 no 5 pp 836ndash839 2018

[16] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014

[17] X Zhao X Liang S Li and B Ai ldquoTwo-cylinder and multi-ring GBSSM for realizing and modeling of vehicle-to-vehiclewideband MIMO channelsrdquo IEEE Transactions on IntelligentTransportation Systems vol 17 no 10 pp 2787ndash2799 2016

[18] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA novel 3Dnonstationary channel model based on the von Mises-Fisherscattering distributionrdquo Mobile Information Systemsvol 2016 Article ID 2161460 pp 1ndash9 2016

[19] Y Yuan C-X Wang Y He M M Alwakeel andE-H M Aggoune ldquo3D wideband non-stationary geometry-based stochastic models for non-isotropic MIMO vehicle-to-vehicle channelsrdquo IEEE Transactions on Wireless Communi-cations vol 14 no 12 pp 6883ndash6895 2015

[20] N Avazov S M R Islam D Park and K S Kwak ldquoStatisticalcharacterization of a 3-D propagation model for V2Vchannels in rectangular tunnelsrdquo IEEE Antennas andWirelessPropagation Letters vol 16 pp 2392ndash2395 2017

[21] H Jiang Z Zhang J Dang and L Wu ldquoA novel 3-D massiveMIMO channel model for vehicle-to-vehicle communicationenvironmentsrdquo IEEE Transactions on Communicationsvol 66 no 1 pp 79ndash90 2018

[22] N Avazov S M R Islam D Park and K S Kwak ldquoCluster-based non-stationary channel modeling for vehicle-to-vehiclecommunicationsrdquo IEEE Antennas and Wireless PropagationLetters vol 16 pp 1419ndash1422 2016

International Journal of Antennas and Propagation 11

[23] D Du X Zeng X Jian L Miao and H Wang ldquoree-di-mensional vehicle-to-vehicle channel modeling with multiplemoving scatterersrdquo Mobile Information Systems vol 2017Article ID 7231417 pp 1ndash14 2017

[24] X LiangW Cao and X Zhao ldquoDoppler power spectra for 3Dvehicle-to-vehicle channels with moving scatterersrdquo IEEEAccess vol 6 pp 42822ndash42828 2018

[25] X Chen Y Fang Y Sun Y Pan and W Xiang ldquoStatisticalcharacterization of novel 3D cluster-based MIMO vehicle-to-vehicle models for urban street scattering environmentsrdquoInternational Journal of Antennas and Propagation vol 2018Article ID 6742346 pp 1ndash11 2018

[26] Q Zhu H Li Y Fu et al ldquoA novel 3D non-stationary wirelessMIMO channel simulator and hardware emulatorrdquo IEEETransactions on Communications vol 66 no 9 pp 3865ndash3878 2018

[27] K Jiang X Chen Q Zhu L Chen D Xu and B Chen ldquoAnovel simulation model for nonstationary rice fading chan-nelsrdquo Wireless Communications and Mobile Computingvol 2018 Article ID 8086073 pp 1ndash9 2018

[28] M Patzold C A Gutierrez and N Youssef ldquoOn the con-sistency of non-stationary multipath fading channels withrespect to the average doppler shift and the doppler spreadrdquoin Proceedings of the IEEE Wireless Communications andNetworking Conference (WCNC) pp 1ndash6 San Francisco CAUSA 2017

[29] Q Zhu W Li C-X Wang et al ldquoTemporal correlations for anon-stationary vehicle-to-vehicle channel model allowingvelocity variationsrdquo IEEE Communications Letters vol 23no 7 pp 1280ndash1284 2019

[30] Q Zhu Y Yang X Chen et al ldquoA novel 3D non-stationaryvehicle-to-vehicle channel model and its spatial-temporalcorrelation propertiesrdquo IEEE Access vol 6 pp 43633ndash436432018

[31] W Dahech M Patzold and N Youssef ldquoA non-stationarymobile-to-mobile multipath fading channel model takingaccount of velocity variations of the mobile stationsrdquo inProceedings of the IEEE International Symposium on Antennasand Propagation pp 1ndash4 Lisbon Portugal April 2015

[32] W Dahech M Patzold C A Gutierrez and N Youssef ldquoAnon-stationary mobile-to-mobile channel model allowing forvelocity and trajectory variations of the mobile stationsrdquo IEEETransactions on Wireless Communications vol 16 no 3pp 1987ndash2000 2017

[33] C A Gutierrez M Patzold W Dahech and N Youssef ldquoAnon-WSSUS mobile-to-mobile channel model assuming ve-locity variations of the mobile stationsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 1ndash6 San Francisco CA USA March 2017

[34] Q Zhu Y Yang C X Wang et al ldquoSpatial correlations of a3D non-stationary MIMO channel model with 3D antennaarrays and 3D arbitrary trajectoriesrdquo IEEE Wireless Com-munications Letters vol 8 no 2 pp 512ndash515 2019

[35] Y Yang Q Zhu W Li et al ldquoA general 3D non-stationarytwin-cluster model for vehicle-to-vehicle MIMO channelsrdquo inProceedings of the Wireless Communications and SignalProcessing pp 1ndash6 Hangzhou China Octobor 2018

[36] H Hofstertter A F Molisch and N Czink ldquoA twin-clusterMIMO channel modelrdquo in Proceedings of the Antennas andPropagation (EuCAP) pp 1ndash8 Nice France November 2006

[37] I Sen and D W Matolak ldquoVehicle-vehicle channel modelsfor the 5-GHz bandrdquo IEEE Transactions on IntelligentTransportation Systems vol 9 no 2 pp 235ndash245 2008

[38] T Zwick C Fischer D Didascalou and W Wiesbeck ldquoAstochastic spatial channel model based on wave-propagationmodelingrdquo IEEE Journal on Selected Areas in Communica-tions vol 18 no 1 pp 6ndash15 2000

[39] K Mammasis and RW Stewart ldquoe Fisher-Bingham spatialcorrelation model for multielement antenna systemsrdquo IEEETransactions on Vehicular Technology vol 58 no 5pp 2130ndash2136 2009

[40] M Konstantinos S W Robert and J S ompson ldquoSpatialfading correlation model using mixtures of Von Mises Fisherdistributionsrdquo IEEE Transactions on Wireless Communica-tions vol 8 no 4 pp 2046ndash2055 2009

[41] A Fayziyev M Patzold E Masson Y Cocheril andM Berbineau ldquoA measurement-based channel model forvehicular communications in tunnelsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 116ndash121 Istanbul Turkey April 2014

[42] S Payami and F Tufvesson ldquoChannel measurements andanalysis for very large array systems at 26 GHzrdquo in Pro-ceedings of the Antennas and Propagation (EUCAP)pp 433ndash437 Prague Czech Republic March 2012

12 International Journal of Antennas and Propagation

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1113957Ri

n(tΔt) κ

4π sinh(κ)1113946Δθ

minus Δθ1113946Δϕ

minus Δϕ

cos θi

n(t) + ζ1113874 1113875eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)( 1113857cos(υ)+sin θi

n(t)+ζ( 1113857sinθi

n(t)( 11138571113970

middot

cos θi

n(t + Δt) + ζ1113874 1113875eκ cos θi

n(t+Δt)+ζ( 1113857cosθi

n(t+Δt)cos(υ)+sin θi

n(t+Δt)+ζ( 1113857sinθi

n(t+Δt)( 11138571113970

middot eminus jk1113938t+Δt

tviSi

n (t)

cos ϕ

iSin

v (t)minus ϕi

n(t)+υ( 11138571113872 1113873cos θi

n(t)+ζ( 1113857cosθiSi

nv (t)+sin θ

i

n(t)+ζ( 1113857sinθiSi

nv (t)1113960 1113961dt

dυdζ

(38)

1113957Ri

n(tΔt) κ

4π sinh(κ)1113946Δθ

minus Δθ1113946Δϕ

minus Δϕ

cos θi

n(t) + ζ1113874 1113875eκ cos θi

n(t)+ζ( 1113857cos θi

n(t)( 1113857cos(υ)+sin θi

n(t)+ζ( 1113857sinθi

n(t)( 11138571113970

middot

cos θi

n(t + Δt) + ζ1113874 1113875eκ cos θi

n(t+Δt)+ζ( 1113857cosθi

n(t+Δt)cos(υ)+sin θi

n(t+Δt)+ζ( 1113857sinθi

n(t+Δt)( 11138571113970

middot eminus jk viSin (t)

cos Ai(tΔtυζ)( )Di(tΔt)+cos Bi(tΔtυζ)( )Ei(tΔt)+sin Ci(tΔtζ)( )Fi(tΔt)[ ]dυdζ

(39)

Finally by substituting (33)ndash(35) into (22) the finalresult of SCF in our proposed model can be obtained Due totime-variant communication environments SCF is alsotime-dependent

42 Time-Variant TCFs e TCF is reduced from STCFwhen Δd is set to be zero en TCF can be derived as(36) It also equals to the product of TCFs at the MT andMR 1113957R

i

n(tΔt) i isin MTMR when the clusters SMTn and

SMRn are independent For simplicity we assumed that theantenna array is placed at the origin of the coordinate

system By substituting (5) and (7) into (36) it can beobtained as (37)

By setting θin(t) θ

i

n(t) + ζ and ϕin(t) ϕi

n(t) + υ andsubstituting them into (37) it yields (38) where ϕiSi

nv (t) and

θiSin

v (t) mean the azimuth and elevation angles of the relativevelocity between i and cluster Si

n respectively It is rea-sonable to assume that the elevation AoDs and azimuthAoDs change linearly during the short time interval asϕi

n(t) k1t + b1 θi

n(t) k2t + b2 where k1 (ϕi

n(t + Δtmax) minus ϕi

n(t))Δtmax k2 (θi

n(t + Δtmax) minus θi

n(t))Δtmaxb1 ϕi

n(t) minus k1t and b2 θi

n(t) minus k2t By integrating theDoppler frequency we can rewrite (38) as (39) where

Ai(tΔt υ ζ)

t + Δt2

1113874 1113875 minus k1 + k2( 1113857 + ϕiSin

v (t) minus b1 + b2 minus υ + ζ

Bi(tΔt υ ζ)

t + Δt2

1113874 1113875 minus k1 minus k2( 1113857 + ϕiSin

v (t) minus b1 minus b2 minus υ minus ζ

Ci(tΔt ζ) k2

t + Δt2

1113874 1113875 + b2 + ζ

Di(tΔt) cos θiSi

n

v (t)sinΔt minus k1 + k2( 11138572( 1113857

minus k1 + k2( 11138571113888 1113889

Ei(tΔt) cos θiSi

n

v (t)sinΔt k1 + k2( 11138572( 1113857

k1 + k2( 11138571113888 1113889

Fi(tΔt) 2sin θiSi

n

v (t)sink2Δt2( 1113857

k21113888 1113889

(40)

1113957Ri

n(tΔt) asympκeκ

4π sinh(κ)

cos θi

n(t)1113874 1113875cos θi

n(t + Δt)1113874 1113875

1113970

eminus jkGi(tΔt)

middot 1113946Δθ

minus Δθeminus κζ22cos kI

i(tΔt)ζ1113872 1113873dζ middot 1113946

Δϕ

minus Δϕeminus Ji(tΔt)υ2 cos kH

i(tΔt)υ1113872 1113873dυ

(41)

International Journal of Antennas and Propagation 7

Measurement data have revealed that small angle spreadsexist under some scenarios In other words κ is usually bigor ζ and υ are small Holding this condition we have

cos(υ minus ζ) asymp 1 cos(υ + ζ) asymp 1 sin(υ minus ζ) asymp υ minus ζ sin(υ + ζ)

asymp υ + ζ cos(ζ) asymp 1 cos(υ) asymp 1 sin(ζ) asymp ζ and sin(υ) asymp υus we can approximate (39) as (41) where

Gi(tΔt) viSi

n (t)

middot Di(tΔt)cos A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)cos B

i(tΔt)1113872 1113873 + viSi

n (t)

middot Fi(tΔt)sin C

i(tΔt)1113872 1113873

Hi(tΔt) viSi

n (t)

middot Di(tΔt)sin A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)sin B

i(tΔt)1113872 1113873

Ii(tΔt) minus viSi

n (t)

middot Di(tΔt)sin A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)sin B

i(tΔt)1113872 1113873 + viSi

n (t)

middot Fi(tΔt)cos C

i(tΔt)1113872 1113873

Ji(tΔt)

κ cos2 θi

n(t)1113874 1113875 + cos2 θi

n(t + Δt)1113874 11138751113874 1113875

4

(42)

1113957Ri

n(tΔt) minusκ

8 sinh(κ)

cos θi

n(t)1113874 1113875cos θi

n(t + Δt)1113874 1113875

1113970

eκminus k2Ii2(tΔt)2κ( 1113857minus k2Hi2(tΔt)4Ji(tΔt)( 1113857minus jkGi(tΔt)

middotIm erfi kHi(tΔt) minus 2jJi(tΔt)Δϕ2

Ji(tΔt)

11139681113872 11138731113872 1113873

Ji(tΔt)

1113968Im erfi kIi(tΔt) minus jκΔθ

2κ

radic( 1113857( 1113857( 1113857

2κ

radic

(43)

where Ai(tΔt) Bi(tΔt) and Ci(tΔt) are the value ofAi(tΔt υ ζ) Bi(tΔt υ ζ) and Ci(tΔt ζ) when υ ζ 0respectively Using the integration formula in (32) theclosed-form expression of (41) can be expressed as (43) Onthis basis we can obtain the final TCF expression of theproposed model

43 Time-Variant DPSDs e DPSD is the Fourier trans-form of TCF 1113957Rusn(tΔt) For the nonstationary aspect of ourproposed model the DPSD can be calculated with short-time Fourier transform as

Sn(f t) 1113946infin

minus infin1113957Rusn(tΔt) eminus j2πfΔt](t minus Δt)dΔt (44)

where window function ](t minus Δt) lasting time is shorter thanthe stationary interval (about several milliseconds [10])

5 Simulation Results and Validation

Firstly in order to evaluate the accuracy of our derived SCFand TCF we calculate the maximum difference between theapproximated and numerical integral results overϕi

n isin [minus 180deg 180deg] and θi

n isin [minus 10deg 10deg] Figure 3 shows theresults of maximum absolute error with different κ nor-malized space lags and time lags As we can see that themaximum absolute error of SCF is less than 0025 when κ ismore than 50 and dλ is less than 3 and it increases as κdecreases or dλ increases Meanwhile the absolute error ofTCF is less than 002 when κ is more than 50 and the time lagis less than 005 s and it increases as κ decreases or the timelag increases Overall the maximum error is acceptable andthus the derived expressions can be used to calculate theSCF and TCF efficiently

Secondly in order to verify the generality of the proposedchannel model ree V2V communication scenarios withdifferent trajectories are selected and given in Figure 4 In thefigure theMTtravels straightly with the fixed velocity and theMR travels in three different velocities with the same de-parture and arrival points Path I has fixed speed and traveldirection and Path II allows speed variation with 2D directionmovement Furthermore Path III has 3D variations in bothvertical and horizontal directions For demonstration pur-pose the distribution of clusters is uniform and we adoptVMF distribution to depict AoAs and AoDs and set κ as395 [39] With the referred data in [41] we assume thatthe speeds of cluster are Gaussian-distributed with 1 kmhmean value and 01 variance Furthermore the carrier fre-quency fc 24GHz ϕS

MTnv ϕS

MRnv θS

MTnv and θS

MRnv are all uni-

formly distributed e former pair distributes over 0 and 2πbut the later one only distributes over minus π36 and π36

It should be mentioned that the V2V channel models in[11ndash26 28 30] only considered fixed velocities of two ter-minals like Path I while the models in [31ndash33] allowed for2D curve trajectories like Path II e absolute values of thetheoretical SCFs of different models including the modelconsidering Path I in [30] and the one focusing on Path II in[32] are simulated and compared in Figure 5 e goodagreement of SCFs between our proposed and other modelsindicates the generalization and compatibility of our pro-posed model In particular the models in [30 32] can beviewed as two special applications in straight or curvedtrajectories

irdly in order to verify the correctness of statisticalproperties of our proposed model under 3D trajectories likePath III the theoretical approximated and simulated valuesof SCFs and TCFs are shown in Figure 6 e x-axis ofFigure 6(a) represents normalized space between antennas

8 International Journal of Antennas and Propagation

320

140

005

120

001

1100 80

0015

60

002

40 200

Max

imum

abso

lute

erro

r

Shape factor ()Normalized antenna

spacing (Δdλ)

(a)

00025

005 200150

10050

Shape factor (κ)Time lag (∆t (s))

0

0005

001

0015

002

Max

imum

abso

lute

erro

r

(b)

Figure 3 Maximum absolute errors between the theoretical and approximate (a) SCFs and (b) TCFs

Start

EndEnd

Start

Path I (1D straight)

Path III (3D arbitrary)

Path II (2D curve)

MTMR

50100 0 ndash50 ndash100 ndash1500

50100

Postion y(m)

Postion x(m)

Posti

on z

(m)

0

1

4

3

5

2

Figure 4 ree typical trajectories of the MR

0

02

04

06

08

1

Model in [28]Model in [30]Proposed model

t = 4st = 8s

t = 0s

Normalized antenna space (Δdλ)0 05 1 15 2

Abso

lute

val

ue o

f SCF

(a)

02

04

06

08

1

Model in [30]Proposed model

00

05 1 15 2

t = 4s

t = 8s

t = 0s

Abso

lute

val

ue o

f SCF

Normalized antenna space (Δdλ)

(b)

Figure 5 Comparison of absolute SCFs for (a) Path I and (b) Path II

International Journal of Antennas and Propagation 9

which is similar at two terminals in the communicationsystem By substituting u1 s1 1 u2 s2 2 and n 1into (21) and (33) we can get the theoretical and approx-imated values of SCFs In addition the measured results in[42] are also shown in Figure 6(a) e good agreement ofSCFs shows the correctness of both the theoretical modeland derivations Similarly by substituting u s 1 and n

1 into (36) and (43) we obtain and show the theoretical andapproximated values of TCFs in Figure 6(b) e goodagreements between theoretical simulated and approxi-mated results reveal that our simulation and derivation arecorrect

Finally substituting TCFs into (44) we can get thetheoretical and simulated values of DPSDs for Path IIIwhich are also shown in Figure 7 It clearly shows thatmoving clusters and two terminals have an influence on thedrifting of DPSDs Moreover good agreements between thesimulated and theoretical results indicate that our proposedmodel is correct

6 Conclusion

is paper has proposed a general channel model charac-terized by 3D scattering environments 3D movements and

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)Measurement

0 05 1 15 2

t = 4s

t = 8s

t = 0s

Normalized antenna space (Δdλ)

Abso

lute

val

ue o

f SCF

(a)

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

0 0005 001 0015 002 0025 003 0035 004

Abso

lute

val

ue o

f TCF

Time lag (Δt)

t = 8s

(b)

Figure 6 Comparison of absolute theoretical and simulated (a) SCFs and (b) TCFs at different time instants

86

4ndash5

0

ndash1000

5

2

10

ndash500

15

0 500 01000Frequency f (HZ)

DPS

D

Time t (s)

(a)

0

5

10

15

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

t = 8s

ndash600 ndash400 ndash200 0 200 400 600Frequency f (HZ)

DPS

D

(b)

Figure 7 (a) eoretical 3D DPSD and (b) theoretical and simulated DPSDs at different time instants for Path III

10 International Journal of Antennas and Propagation

3D-shaped antenna arrays of two terminals for V2V com-munication system e upgraded algorithms for time-evolving parameters such as the number of paths pathdelays path powers and angles have been developed andanalyzed in detail Meanwhile the approximated expressionsof statistical properties including SCF TCF and DPSD havebeen derived and verified by simulations Simulated andanalyzed results show that different moving trajectoriessignificantly affect the V2V channel characteristic Mean-while the generalization and compatibility of our proposedmodel also demonstrate that some previous models with 2Dor even 1D movements have special applications e newmodel can be used to develop analyze and test realistic V2Vcommunication systems in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Key ScientificInstrument and Equipment Development Project (Grant no61827801) Aeronautical Science Foundation of China(Grant no 2017ZC52021) and Open Foundation forGraduate Innovation of NUAA (Grant no KFJJ 20180408)

References

[1] C-X Wang X Cheng and D I Laurenson ldquoVehicle-to-vehicle channel modeling andmeasurements recent advancesand future challengesrdquo IEEE Communications Magazinevol 47 no 11 pp 96ndash103 2009

[2] K Guan B Ai M Liso Nicolas et al ldquoOn the influence ofscattering from traffic signs in vehicle-to-x communicationsrdquoIEEE Transactions on Vehicular Technology vol 65 no 8pp 5835ndash5849 2016

[3] X Cheng C-X Wang B Ai and H Aggoune ldquoEnvelopelevel crossing rate and average fade duration of nonisotropicvehicle-to-vehicle ricean fading channelsrdquo IEEE Transactionson Intelligent Transportation Systems vol 15 no 1 pp 62ndash722014

[4] Q Zhu C Xue X Chen and Y Yang ldquoA newMIMO channelmodel incorporating antenna effectsrdquo Progress In Electro-magnetics Research M vol 50 pp 129ndash140 2016

[5] A G Zajic and G L Stubber ldquoSpace-time correlated mobile-to-mobile channels modelling and simulationrdquo IEEETransactions on Vehicular Technology vol 57 no 2pp 715ndash726 2008

[6] W Fan X Carreno Bautista de Lisbona F Sun J O NielsenM B Knudsen and G F Pedersen ldquoEmulating spatialcharacteristics of MIMO channels for OTA testingrdquo IEEETransactions on Antennas and Propagation vol 61 no 8pp 4306ndash4314 2013

[7] J Zhang Y Zhang Y Yu R Xu Q Zheng and P Zhangldquo3-D MIMO how much does it meet our expectationsobserved from channel measurementsrdquo IEEE Journal on

Selected Areas in Communications vol 35 no 8pp 1887ndash1903 2017

[8] J Karedal F Tufvesson N Czink et al ldquoA geometry-basedstochastic MIMO model for vehicle-to-vehicle communica-tionsrdquo IEEE Transactions onWireless Communications vol 8no 7 pp 3646ndash3657 2009

[9] A Paier T Zemen L Bernado et al ldquoNon-WSSUS vehicularchannel characterization in high-way and urban scenarios at52 GHz using the local scattering functionrdquo in Proceedings ofthe 2008 International ITG Workshop on Smart Antennaspp 9ndash15 Darmstadt Germany February 2008

[10] R He O Renaudin V-M Kolmonen et al ldquoCharacterizationof quasi-stationarity regions for vehicle-to-vehicle radiochannelsrdquo IEEE Transactions on Antennas and Propagationvol 63 no 5 pp 2237ndash2251 2015

[11] YMa L Yang andX Zheng ldquoA geometry-based non-stationaryMIMO channel model for vehicular communicationsrdquo ChinaCommunications vol 15 no 7 pp 30ndash38 2018

[12] C A Gutirrez J T Gutirrez-Mena J M Luna-RiveraD U Campos-Delgado R Velzquez and M Patzold ldquoGe-ometry-based statistical modeling of non-WSSUS mobile-to-mobile Rayleigh fading channelsrdquo IEEE Transactions onVehicular Technology vol 67 no 1 pp 362ndash377 2018

[13] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the Vehicular Technology Confer-ence pp 1ndash6 San Francisco CA USA September 2011

[14] A Borhani and M Patzold ldquoCorrelation and spectralproperties of vehicle-to-vehicle channels in the presence ofmoving scatterersrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 9 pp 4228ndash4239 2013

[15] H Jiang Z Zhang L Wu and J Dang ldquoNovel 3-D irregular-shaped geometry-based channel modeling for semi-ellipsoidvehicle-to-vehicle scattering environmentsrdquo IEEE WirelessCommunications Letters vol 7 no 5 pp 836ndash839 2018

[16] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014

[17] X Zhao X Liang S Li and B Ai ldquoTwo-cylinder and multi-ring GBSSM for realizing and modeling of vehicle-to-vehiclewideband MIMO channelsrdquo IEEE Transactions on IntelligentTransportation Systems vol 17 no 10 pp 2787ndash2799 2016

[18] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA novel 3Dnonstationary channel model based on the von Mises-Fisherscattering distributionrdquo Mobile Information Systemsvol 2016 Article ID 2161460 pp 1ndash9 2016

[19] Y Yuan C-X Wang Y He M M Alwakeel andE-H M Aggoune ldquo3D wideband non-stationary geometry-based stochastic models for non-isotropic MIMO vehicle-to-vehicle channelsrdquo IEEE Transactions on Wireless Communi-cations vol 14 no 12 pp 6883ndash6895 2015

[20] N Avazov S M R Islam D Park and K S Kwak ldquoStatisticalcharacterization of a 3-D propagation model for V2Vchannels in rectangular tunnelsrdquo IEEE Antennas andWirelessPropagation Letters vol 16 pp 2392ndash2395 2017

[21] H Jiang Z Zhang J Dang and L Wu ldquoA novel 3-D massiveMIMO channel model for vehicle-to-vehicle communicationenvironmentsrdquo IEEE Transactions on Communicationsvol 66 no 1 pp 79ndash90 2018

[22] N Avazov S M R Islam D Park and K S Kwak ldquoCluster-based non-stationary channel modeling for vehicle-to-vehiclecommunicationsrdquo IEEE Antennas and Wireless PropagationLetters vol 16 pp 1419ndash1422 2016

International Journal of Antennas and Propagation 11

[23] D Du X Zeng X Jian L Miao and H Wang ldquoree-di-mensional vehicle-to-vehicle channel modeling with multiplemoving scatterersrdquo Mobile Information Systems vol 2017Article ID 7231417 pp 1ndash14 2017

[24] X LiangW Cao and X Zhao ldquoDoppler power spectra for 3Dvehicle-to-vehicle channels with moving scatterersrdquo IEEEAccess vol 6 pp 42822ndash42828 2018

[25] X Chen Y Fang Y Sun Y Pan and W Xiang ldquoStatisticalcharacterization of novel 3D cluster-based MIMO vehicle-to-vehicle models for urban street scattering environmentsrdquoInternational Journal of Antennas and Propagation vol 2018Article ID 6742346 pp 1ndash11 2018

[26] Q Zhu H Li Y Fu et al ldquoA novel 3D non-stationary wirelessMIMO channel simulator and hardware emulatorrdquo IEEETransactions on Communications vol 66 no 9 pp 3865ndash3878 2018

[27] K Jiang X Chen Q Zhu L Chen D Xu and B Chen ldquoAnovel simulation model for nonstationary rice fading chan-nelsrdquo Wireless Communications and Mobile Computingvol 2018 Article ID 8086073 pp 1ndash9 2018

[28] M Patzold C A Gutierrez and N Youssef ldquoOn the con-sistency of non-stationary multipath fading channels withrespect to the average doppler shift and the doppler spreadrdquoin Proceedings of the IEEE Wireless Communications andNetworking Conference (WCNC) pp 1ndash6 San Francisco CAUSA 2017

[29] Q Zhu W Li C-X Wang et al ldquoTemporal correlations for anon-stationary vehicle-to-vehicle channel model allowingvelocity variationsrdquo IEEE Communications Letters vol 23no 7 pp 1280ndash1284 2019

[30] Q Zhu Y Yang X Chen et al ldquoA novel 3D non-stationaryvehicle-to-vehicle channel model and its spatial-temporalcorrelation propertiesrdquo IEEE Access vol 6 pp 43633ndash436432018

[31] W Dahech M Patzold and N Youssef ldquoA non-stationarymobile-to-mobile multipath fading channel model takingaccount of velocity variations of the mobile stationsrdquo inProceedings of the IEEE International Symposium on Antennasand Propagation pp 1ndash4 Lisbon Portugal April 2015

[32] W Dahech M Patzold C A Gutierrez and N Youssef ldquoAnon-stationary mobile-to-mobile channel model allowing forvelocity and trajectory variations of the mobile stationsrdquo IEEETransactions on Wireless Communications vol 16 no 3pp 1987ndash2000 2017

[33] C A Gutierrez M Patzold W Dahech and N Youssef ldquoAnon-WSSUS mobile-to-mobile channel model assuming ve-locity variations of the mobile stationsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 1ndash6 San Francisco CA USA March 2017

[34] Q Zhu Y Yang C X Wang et al ldquoSpatial correlations of a3D non-stationary MIMO channel model with 3D antennaarrays and 3D arbitrary trajectoriesrdquo IEEE Wireless Com-munications Letters vol 8 no 2 pp 512ndash515 2019

[35] Y Yang Q Zhu W Li et al ldquoA general 3D non-stationarytwin-cluster model for vehicle-to-vehicle MIMO channelsrdquo inProceedings of the Wireless Communications and SignalProcessing pp 1ndash6 Hangzhou China Octobor 2018

[36] H Hofstertter A F Molisch and N Czink ldquoA twin-clusterMIMO channel modelrdquo in Proceedings of the Antennas andPropagation (EuCAP) pp 1ndash8 Nice France November 2006

[37] I Sen and D W Matolak ldquoVehicle-vehicle channel modelsfor the 5-GHz bandrdquo IEEE Transactions on IntelligentTransportation Systems vol 9 no 2 pp 235ndash245 2008

[38] T Zwick C Fischer D Didascalou and W Wiesbeck ldquoAstochastic spatial channel model based on wave-propagationmodelingrdquo IEEE Journal on Selected Areas in Communica-tions vol 18 no 1 pp 6ndash15 2000

[39] K Mammasis and RW Stewart ldquoe Fisher-Bingham spatialcorrelation model for multielement antenna systemsrdquo IEEETransactions on Vehicular Technology vol 58 no 5pp 2130ndash2136 2009

[40] M Konstantinos S W Robert and J S ompson ldquoSpatialfading correlation model using mixtures of Von Mises Fisherdistributionsrdquo IEEE Transactions on Wireless Communica-tions vol 8 no 4 pp 2046ndash2055 2009

[41] A Fayziyev M Patzold E Masson Y Cocheril andM Berbineau ldquoA measurement-based channel model forvehicular communications in tunnelsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 116ndash121 Istanbul Turkey April 2014

[42] S Payami and F Tufvesson ldquoChannel measurements andanalysis for very large array systems at 26 GHzrdquo in Pro-ceedings of the Antennas and Propagation (EUCAP)pp 433ndash437 Prague Czech Republic March 2012

12 International Journal of Antennas and Propagation

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Measurement data have revealed that small angle spreadsexist under some scenarios In other words κ is usually bigor ζ and υ are small Holding this condition we have

cos(υ minus ζ) asymp 1 cos(υ + ζ) asymp 1 sin(υ minus ζ) asymp υ minus ζ sin(υ + ζ)

asymp υ + ζ cos(ζ) asymp 1 cos(υ) asymp 1 sin(ζ) asymp ζ and sin(υ) asymp υus we can approximate (39) as (41) where

Gi(tΔt) viSi

n (t)

middot Di(tΔt)cos A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)cos B

i(tΔt)1113872 1113873 + viSi

n (t)

middot Fi(tΔt)sin C

i(tΔt)1113872 1113873

Hi(tΔt) viSi

n (t)

middot Di(tΔt)sin A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)sin B

i(tΔt)1113872 1113873

Ii(tΔt) minus viSi

n (t)

middot Di(tΔt)sin A

i(tΔt)1113872 1113873 + viSi

n (t)

middot Ei(tΔt)sin B

i(tΔt)1113872 1113873 + viSi

n (t)

middot Fi(tΔt)cos C

i(tΔt)1113872 1113873

Ji(tΔt)

κ cos2 θi

n(t)1113874 1113875 + cos2 θi

n(t + Δt)1113874 11138751113874 1113875

4

(42)

1113957Ri

n(tΔt) minusκ

8 sinh(κ)

cos θi

n(t)1113874 1113875cos θi

n(t + Δt)1113874 1113875

1113970

eκminus k2Ii2(tΔt)2κ( 1113857minus k2Hi2(tΔt)4Ji(tΔt)( 1113857minus jkGi(tΔt)

middotIm erfi kHi(tΔt) minus 2jJi(tΔt)Δϕ2

Ji(tΔt)

11139681113872 11138731113872 1113873

Ji(tΔt)

1113968Im erfi kIi(tΔt) minus jκΔθ

2κ

radic( 1113857( 1113857( 1113857

2κ

radic

(43)

where Ai(tΔt) Bi(tΔt) and Ci(tΔt) are the value ofAi(tΔt υ ζ) Bi(tΔt υ ζ) and Ci(tΔt ζ) when υ ζ 0respectively Using the integration formula in (32) theclosed-form expression of (41) can be expressed as (43) Onthis basis we can obtain the final TCF expression of theproposed model

43 Time-Variant DPSDs e DPSD is the Fourier trans-form of TCF 1113957Rusn(tΔt) For the nonstationary aspect of ourproposed model the DPSD can be calculated with short-time Fourier transform as

Sn(f t) 1113946infin

minus infin1113957Rusn(tΔt) eminus j2πfΔt](t minus Δt)dΔt (44)

where window function ](t minus Δt) lasting time is shorter thanthe stationary interval (about several milliseconds [10])

5 Simulation Results and Validation

Firstly in order to evaluate the accuracy of our derived SCFand TCF we calculate the maximum difference between theapproximated and numerical integral results overϕi

n isin [minus 180deg 180deg] and θi

n isin [minus 10deg 10deg] Figure 3 shows theresults of maximum absolute error with different κ nor-malized space lags and time lags As we can see that themaximum absolute error of SCF is less than 0025 when κ ismore than 50 and dλ is less than 3 and it increases as κdecreases or dλ increases Meanwhile the absolute error ofTCF is less than 002 when κ is more than 50 and the time lagis less than 005 s and it increases as κ decreases or the timelag increases Overall the maximum error is acceptable andthus the derived expressions can be used to calculate theSCF and TCF efficiently

Secondly in order to verify the generality of the proposedchannel model ree V2V communication scenarios withdifferent trajectories are selected and given in Figure 4 In thefigure theMTtravels straightly with the fixed velocity and theMR travels in three different velocities with the same de-parture and arrival points Path I has fixed speed and traveldirection and Path II allows speed variation with 2D directionmovement Furthermore Path III has 3D variations in bothvertical and horizontal directions For demonstration pur-pose the distribution of clusters is uniform and we adoptVMF distribution to depict AoAs and AoDs and set κ as395 [39] With the referred data in [41] we assume thatthe speeds of cluster are Gaussian-distributed with 1 kmhmean value and 01 variance Furthermore the carrier fre-quency fc 24GHz ϕS

MTnv ϕS

MRnv θS

MTnv and θS

MRnv are all uni-

formly distributed e former pair distributes over 0 and 2πbut the later one only distributes over minus π36 and π36

It should be mentioned that the V2V channel models in[11ndash26 28 30] only considered fixed velocities of two ter-minals like Path I while the models in [31ndash33] allowed for2D curve trajectories like Path II e absolute values of thetheoretical SCFs of different models including the modelconsidering Path I in [30] and the one focusing on Path II in[32] are simulated and compared in Figure 5 e goodagreement of SCFs between our proposed and other modelsindicates the generalization and compatibility of our pro-posed model In particular the models in [30 32] can beviewed as two special applications in straight or curvedtrajectories

irdly in order to verify the correctness of statisticalproperties of our proposed model under 3D trajectories likePath III the theoretical approximated and simulated valuesof SCFs and TCFs are shown in Figure 6 e x-axis ofFigure 6(a) represents normalized space between antennas

8 International Journal of Antennas and Propagation

320

140

005

120

001

1100 80

0015

60

002

40 200

Max

imum

abso

lute

erro

r

Shape factor ()Normalized antenna

spacing (Δdλ)

(a)

00025

005 200150

10050

Shape factor (κ)Time lag (∆t (s))

0

0005

001

0015

002

Max

imum

abso

lute

erro

r

(b)

Figure 3 Maximum absolute errors between the theoretical and approximate (a) SCFs and (b) TCFs

Start

EndEnd

Start

Path I (1D straight)

Path III (3D arbitrary)

Path II (2D curve)

MTMR

50100 0 ndash50 ndash100 ndash1500

50100

Postion y(m)

Postion x(m)

Posti

on z

(m)

0

1

4

3

5

2

Figure 4 ree typical trajectories of the MR

0

02

04

06

08

1

Model in [28]Model in [30]Proposed model

t = 4st = 8s

t = 0s

Normalized antenna space (Δdλ)0 05 1 15 2

Abso

lute

val

ue o

f SCF

(a)

02

04

06

08

1

Model in [30]Proposed model

00

05 1 15 2

t = 4s

t = 8s

t = 0s

Abso

lute

val

ue o

f SCF

Normalized antenna space (Δdλ)

(b)

Figure 5 Comparison of absolute SCFs for (a) Path I and (b) Path II

International Journal of Antennas and Propagation 9

which is similar at two terminals in the communicationsystem By substituting u1 s1 1 u2 s2 2 and n 1into (21) and (33) we can get the theoretical and approx-imated values of SCFs In addition the measured results in[42] are also shown in Figure 6(a) e good agreement ofSCFs shows the correctness of both the theoretical modeland derivations Similarly by substituting u s 1 and n

1 into (36) and (43) we obtain and show the theoretical andapproximated values of TCFs in Figure 6(b) e goodagreements between theoretical simulated and approxi-mated results reveal that our simulation and derivation arecorrect

Finally substituting TCFs into (44) we can get thetheoretical and simulated values of DPSDs for Path IIIwhich are also shown in Figure 7 It clearly shows thatmoving clusters and two terminals have an influence on thedrifting of DPSDs Moreover good agreements between thesimulated and theoretical results indicate that our proposedmodel is correct

6 Conclusion

is paper has proposed a general channel model charac-terized by 3D scattering environments 3D movements and

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)Measurement

0 05 1 15 2

t = 4s

t = 8s

t = 0s

Normalized antenna space (Δdλ)

Abso

lute

val

ue o

f SCF

(a)

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

0 0005 001 0015 002 0025 003 0035 004

Abso

lute

val

ue o

f TCF

Time lag (Δt)

t = 8s

(b)

Figure 6 Comparison of absolute theoretical and simulated (a) SCFs and (b) TCFs at different time instants

86

4ndash5

0

ndash1000

5

2

10

ndash500

15

0 500 01000Frequency f (HZ)

DPS

D

Time t (s)

(a)

0

5

10

15

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

t = 8s

ndash600 ndash400 ndash200 0 200 400 600Frequency f (HZ)

DPS

D

(b)

Figure 7 (a) eoretical 3D DPSD and (b) theoretical and simulated DPSDs at different time instants for Path III

10 International Journal of Antennas and Propagation

3D-shaped antenna arrays of two terminals for V2V com-munication system e upgraded algorithms for time-evolving parameters such as the number of paths pathdelays path powers and angles have been developed andanalyzed in detail Meanwhile the approximated expressionsof statistical properties including SCF TCF and DPSD havebeen derived and verified by simulations Simulated andanalyzed results show that different moving trajectoriessignificantly affect the V2V channel characteristic Mean-while the generalization and compatibility of our proposedmodel also demonstrate that some previous models with 2Dor even 1D movements have special applications e newmodel can be used to develop analyze and test realistic V2Vcommunication systems in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Key ScientificInstrument and Equipment Development Project (Grant no61827801) Aeronautical Science Foundation of China(Grant no 2017ZC52021) and Open Foundation forGraduate Innovation of NUAA (Grant no KFJJ 20180408)

References

[1] C-X Wang X Cheng and D I Laurenson ldquoVehicle-to-vehicle channel modeling andmeasurements recent advancesand future challengesrdquo IEEE Communications Magazinevol 47 no 11 pp 96ndash103 2009

[2] K Guan B Ai M Liso Nicolas et al ldquoOn the influence ofscattering from traffic signs in vehicle-to-x communicationsrdquoIEEE Transactions on Vehicular Technology vol 65 no 8pp 5835ndash5849 2016

[3] X Cheng C-X Wang B Ai and H Aggoune ldquoEnvelopelevel crossing rate and average fade duration of nonisotropicvehicle-to-vehicle ricean fading channelsrdquo IEEE Transactionson Intelligent Transportation Systems vol 15 no 1 pp 62ndash722014

[4] Q Zhu C Xue X Chen and Y Yang ldquoA newMIMO channelmodel incorporating antenna effectsrdquo Progress In Electro-magnetics Research M vol 50 pp 129ndash140 2016

[5] A G Zajic and G L Stubber ldquoSpace-time correlated mobile-to-mobile channels modelling and simulationrdquo IEEETransactions on Vehicular Technology vol 57 no 2pp 715ndash726 2008

[6] W Fan X Carreno Bautista de Lisbona F Sun J O NielsenM B Knudsen and G F Pedersen ldquoEmulating spatialcharacteristics of MIMO channels for OTA testingrdquo IEEETransactions on Antennas and Propagation vol 61 no 8pp 4306ndash4314 2013

[7] J Zhang Y Zhang Y Yu R Xu Q Zheng and P Zhangldquo3-D MIMO how much does it meet our expectationsobserved from channel measurementsrdquo IEEE Journal on

Selected Areas in Communications vol 35 no 8pp 1887ndash1903 2017

[8] J Karedal F Tufvesson N Czink et al ldquoA geometry-basedstochastic MIMO model for vehicle-to-vehicle communica-tionsrdquo IEEE Transactions onWireless Communications vol 8no 7 pp 3646ndash3657 2009

[9] A Paier T Zemen L Bernado et al ldquoNon-WSSUS vehicularchannel characterization in high-way and urban scenarios at52 GHz using the local scattering functionrdquo in Proceedings ofthe 2008 International ITG Workshop on Smart Antennaspp 9ndash15 Darmstadt Germany February 2008

[10] R He O Renaudin V-M Kolmonen et al ldquoCharacterizationof quasi-stationarity regions for vehicle-to-vehicle radiochannelsrdquo IEEE Transactions on Antennas and Propagationvol 63 no 5 pp 2237ndash2251 2015

[11] YMa L Yang andX Zheng ldquoA geometry-based non-stationaryMIMO channel model for vehicular communicationsrdquo ChinaCommunications vol 15 no 7 pp 30ndash38 2018

[12] C A Gutirrez J T Gutirrez-Mena J M Luna-RiveraD U Campos-Delgado R Velzquez and M Patzold ldquoGe-ometry-based statistical modeling of non-WSSUS mobile-to-mobile Rayleigh fading channelsrdquo IEEE Transactions onVehicular Technology vol 67 no 1 pp 362ndash377 2018

[13] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the Vehicular Technology Confer-ence pp 1ndash6 San Francisco CA USA September 2011

[14] A Borhani and M Patzold ldquoCorrelation and spectralproperties of vehicle-to-vehicle channels in the presence ofmoving scatterersrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 9 pp 4228ndash4239 2013

[15] H Jiang Z Zhang L Wu and J Dang ldquoNovel 3-D irregular-shaped geometry-based channel modeling for semi-ellipsoidvehicle-to-vehicle scattering environmentsrdquo IEEE WirelessCommunications Letters vol 7 no 5 pp 836ndash839 2018

[16] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014

[17] X Zhao X Liang S Li and B Ai ldquoTwo-cylinder and multi-ring GBSSM for realizing and modeling of vehicle-to-vehiclewideband MIMO channelsrdquo IEEE Transactions on IntelligentTransportation Systems vol 17 no 10 pp 2787ndash2799 2016

[18] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA novel 3Dnonstationary channel model based on the von Mises-Fisherscattering distributionrdquo Mobile Information Systemsvol 2016 Article ID 2161460 pp 1ndash9 2016

[19] Y Yuan C-X Wang Y He M M Alwakeel andE-H M Aggoune ldquo3D wideband non-stationary geometry-based stochastic models for non-isotropic MIMO vehicle-to-vehicle channelsrdquo IEEE Transactions on Wireless Communi-cations vol 14 no 12 pp 6883ndash6895 2015

[20] N Avazov S M R Islam D Park and K S Kwak ldquoStatisticalcharacterization of a 3-D propagation model for V2Vchannels in rectangular tunnelsrdquo IEEE Antennas andWirelessPropagation Letters vol 16 pp 2392ndash2395 2017

[21] H Jiang Z Zhang J Dang and L Wu ldquoA novel 3-D massiveMIMO channel model for vehicle-to-vehicle communicationenvironmentsrdquo IEEE Transactions on Communicationsvol 66 no 1 pp 79ndash90 2018

[22] N Avazov S M R Islam D Park and K S Kwak ldquoCluster-based non-stationary channel modeling for vehicle-to-vehiclecommunicationsrdquo IEEE Antennas and Wireless PropagationLetters vol 16 pp 1419ndash1422 2016

International Journal of Antennas and Propagation 11

[23] D Du X Zeng X Jian L Miao and H Wang ldquoree-di-mensional vehicle-to-vehicle channel modeling with multiplemoving scatterersrdquo Mobile Information Systems vol 2017Article ID 7231417 pp 1ndash14 2017

[24] X LiangW Cao and X Zhao ldquoDoppler power spectra for 3Dvehicle-to-vehicle channels with moving scatterersrdquo IEEEAccess vol 6 pp 42822ndash42828 2018

[25] X Chen Y Fang Y Sun Y Pan and W Xiang ldquoStatisticalcharacterization of novel 3D cluster-based MIMO vehicle-to-vehicle models for urban street scattering environmentsrdquoInternational Journal of Antennas and Propagation vol 2018Article ID 6742346 pp 1ndash11 2018

[26] Q Zhu H Li Y Fu et al ldquoA novel 3D non-stationary wirelessMIMO channel simulator and hardware emulatorrdquo IEEETransactions on Communications vol 66 no 9 pp 3865ndash3878 2018

[27] K Jiang X Chen Q Zhu L Chen D Xu and B Chen ldquoAnovel simulation model for nonstationary rice fading chan-nelsrdquo Wireless Communications and Mobile Computingvol 2018 Article ID 8086073 pp 1ndash9 2018

[28] M Patzold C A Gutierrez and N Youssef ldquoOn the con-sistency of non-stationary multipath fading channels withrespect to the average doppler shift and the doppler spreadrdquoin Proceedings of the IEEE Wireless Communications andNetworking Conference (WCNC) pp 1ndash6 San Francisco CAUSA 2017

[29] Q Zhu W Li C-X Wang et al ldquoTemporal correlations for anon-stationary vehicle-to-vehicle channel model allowingvelocity variationsrdquo IEEE Communications Letters vol 23no 7 pp 1280ndash1284 2019

[30] Q Zhu Y Yang X Chen et al ldquoA novel 3D non-stationaryvehicle-to-vehicle channel model and its spatial-temporalcorrelation propertiesrdquo IEEE Access vol 6 pp 43633ndash436432018

[31] W Dahech M Patzold and N Youssef ldquoA non-stationarymobile-to-mobile multipath fading channel model takingaccount of velocity variations of the mobile stationsrdquo inProceedings of the IEEE International Symposium on Antennasand Propagation pp 1ndash4 Lisbon Portugal April 2015

[32] W Dahech M Patzold C A Gutierrez and N Youssef ldquoAnon-stationary mobile-to-mobile channel model allowing forvelocity and trajectory variations of the mobile stationsrdquo IEEETransactions on Wireless Communications vol 16 no 3pp 1987ndash2000 2017

[33] C A Gutierrez M Patzold W Dahech and N Youssef ldquoAnon-WSSUS mobile-to-mobile channel model assuming ve-locity variations of the mobile stationsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 1ndash6 San Francisco CA USA March 2017

[34] Q Zhu Y Yang C X Wang et al ldquoSpatial correlations of a3D non-stationary MIMO channel model with 3D antennaarrays and 3D arbitrary trajectoriesrdquo IEEE Wireless Com-munications Letters vol 8 no 2 pp 512ndash515 2019

[35] Y Yang Q Zhu W Li et al ldquoA general 3D non-stationarytwin-cluster model for vehicle-to-vehicle MIMO channelsrdquo inProceedings of the Wireless Communications and SignalProcessing pp 1ndash6 Hangzhou China Octobor 2018

[36] H Hofstertter A F Molisch and N Czink ldquoA twin-clusterMIMO channel modelrdquo in Proceedings of the Antennas andPropagation (EuCAP) pp 1ndash8 Nice France November 2006

[37] I Sen and D W Matolak ldquoVehicle-vehicle channel modelsfor the 5-GHz bandrdquo IEEE Transactions on IntelligentTransportation Systems vol 9 no 2 pp 235ndash245 2008

[38] T Zwick C Fischer D Didascalou and W Wiesbeck ldquoAstochastic spatial channel model based on wave-propagationmodelingrdquo IEEE Journal on Selected Areas in Communica-tions vol 18 no 1 pp 6ndash15 2000

[39] K Mammasis and RW Stewart ldquoe Fisher-Bingham spatialcorrelation model for multielement antenna systemsrdquo IEEETransactions on Vehicular Technology vol 58 no 5pp 2130ndash2136 2009

[40] M Konstantinos S W Robert and J S ompson ldquoSpatialfading correlation model using mixtures of Von Mises Fisherdistributionsrdquo IEEE Transactions on Wireless Communica-tions vol 8 no 4 pp 2046ndash2055 2009

[41] A Fayziyev M Patzold E Masson Y Cocheril andM Berbineau ldquoA measurement-based channel model forvehicular communications in tunnelsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 116ndash121 Istanbul Turkey April 2014

[42] S Payami and F Tufvesson ldquoChannel measurements andanalysis for very large array systems at 26 GHzrdquo in Pro-ceedings of the Antennas and Propagation (EUCAP)pp 433ndash437 Prague Czech Republic March 2012

12 International Journal of Antennas and Propagation

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

320

140

005

120

001

1100 80

0015

60

002

40 200

Max

imum

abso

lute

erro

r

Shape factor ()Normalized antenna

spacing (Δdλ)

(a)

00025

005 200150

10050

Shape factor (κ)Time lag (∆t (s))

0

0005

001

0015

002

Max

imum

abso

lute

erro

r

(b)

Figure 3 Maximum absolute errors between the theoretical and approximate (a) SCFs and (b) TCFs

Start

EndEnd

Start

Path I (1D straight)

Path III (3D arbitrary)

Path II (2D curve)

MTMR

50100 0 ndash50 ndash100 ndash1500

50100

Postion y(m)

Postion x(m)

Posti

on z

(m)

0

1

4

3

5

2

Figure 4 ree typical trajectories of the MR

0

02

04

06

08

1

Model in [28]Model in [30]Proposed model

t = 4st = 8s

t = 0s

Normalized antenna space (Δdλ)0 05 1 15 2

Abso

lute

val

ue o

f SCF

(a)

02

04

06

08

1

Model in [30]Proposed model

00

05 1 15 2

t = 4s

t = 8s

t = 0s

Abso

lute

val

ue o

f SCF

Normalized antenna space (Δdλ)

(b)

Figure 5 Comparison of absolute SCFs for (a) Path I and (b) Path II

International Journal of Antennas and Propagation 9

which is similar at two terminals in the communicationsystem By substituting u1 s1 1 u2 s2 2 and n 1into (21) and (33) we can get the theoretical and approx-imated values of SCFs In addition the measured results in[42] are also shown in Figure 6(a) e good agreement ofSCFs shows the correctness of both the theoretical modeland derivations Similarly by substituting u s 1 and n

1 into (36) and (43) we obtain and show the theoretical andapproximated values of TCFs in Figure 6(b) e goodagreements between theoretical simulated and approxi-mated results reveal that our simulation and derivation arecorrect

Finally substituting TCFs into (44) we can get thetheoretical and simulated values of DPSDs for Path IIIwhich are also shown in Figure 7 It clearly shows thatmoving clusters and two terminals have an influence on thedrifting of DPSDs Moreover good agreements between thesimulated and theoretical results indicate that our proposedmodel is correct

6 Conclusion

is paper has proposed a general channel model charac-terized by 3D scattering environments 3D movements and

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)Measurement

0 05 1 15 2

t = 4s

t = 8s

t = 0s

Normalized antenna space (Δdλ)

Abso

lute

val

ue o

f SCF

(a)

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

0 0005 001 0015 002 0025 003 0035 004

Abso

lute

val

ue o

f TCF

Time lag (Δt)

t = 8s

(b)

Figure 6 Comparison of absolute theoretical and simulated (a) SCFs and (b) TCFs at different time instants

86

4ndash5

0

ndash1000

5

2

10

ndash500

15

0 500 01000Frequency f (HZ)

DPS

D

Time t (s)

(a)

0

5

10

15

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

t = 8s

ndash600 ndash400 ndash200 0 200 400 600Frequency f (HZ)

DPS

D

(b)

Figure 7 (a) eoretical 3D DPSD and (b) theoretical and simulated DPSDs at different time instants for Path III

10 International Journal of Antennas and Propagation

3D-shaped antenna arrays of two terminals for V2V com-munication system e upgraded algorithms for time-evolving parameters such as the number of paths pathdelays path powers and angles have been developed andanalyzed in detail Meanwhile the approximated expressionsof statistical properties including SCF TCF and DPSD havebeen derived and verified by simulations Simulated andanalyzed results show that different moving trajectoriessignificantly affect the V2V channel characteristic Mean-while the generalization and compatibility of our proposedmodel also demonstrate that some previous models with 2Dor even 1D movements have special applications e newmodel can be used to develop analyze and test realistic V2Vcommunication systems in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Key ScientificInstrument and Equipment Development Project (Grant no61827801) Aeronautical Science Foundation of China(Grant no 2017ZC52021) and Open Foundation forGraduate Innovation of NUAA (Grant no KFJJ 20180408)

References

[1] C-X Wang X Cheng and D I Laurenson ldquoVehicle-to-vehicle channel modeling andmeasurements recent advancesand future challengesrdquo IEEE Communications Magazinevol 47 no 11 pp 96ndash103 2009

[2] K Guan B Ai M Liso Nicolas et al ldquoOn the influence ofscattering from traffic signs in vehicle-to-x communicationsrdquoIEEE Transactions on Vehicular Technology vol 65 no 8pp 5835ndash5849 2016

[3] X Cheng C-X Wang B Ai and H Aggoune ldquoEnvelopelevel crossing rate and average fade duration of nonisotropicvehicle-to-vehicle ricean fading channelsrdquo IEEE Transactionson Intelligent Transportation Systems vol 15 no 1 pp 62ndash722014

[4] Q Zhu C Xue X Chen and Y Yang ldquoA newMIMO channelmodel incorporating antenna effectsrdquo Progress In Electro-magnetics Research M vol 50 pp 129ndash140 2016

[5] A G Zajic and G L Stubber ldquoSpace-time correlated mobile-to-mobile channels modelling and simulationrdquo IEEETransactions on Vehicular Technology vol 57 no 2pp 715ndash726 2008

[6] W Fan X Carreno Bautista de Lisbona F Sun J O NielsenM B Knudsen and G F Pedersen ldquoEmulating spatialcharacteristics of MIMO channels for OTA testingrdquo IEEETransactions on Antennas and Propagation vol 61 no 8pp 4306ndash4314 2013

[7] J Zhang Y Zhang Y Yu R Xu Q Zheng and P Zhangldquo3-D MIMO how much does it meet our expectationsobserved from channel measurementsrdquo IEEE Journal on

Selected Areas in Communications vol 35 no 8pp 1887ndash1903 2017

[8] J Karedal F Tufvesson N Czink et al ldquoA geometry-basedstochastic MIMO model for vehicle-to-vehicle communica-tionsrdquo IEEE Transactions onWireless Communications vol 8no 7 pp 3646ndash3657 2009

[9] A Paier T Zemen L Bernado et al ldquoNon-WSSUS vehicularchannel characterization in high-way and urban scenarios at52 GHz using the local scattering functionrdquo in Proceedings ofthe 2008 International ITG Workshop on Smart Antennaspp 9ndash15 Darmstadt Germany February 2008

[10] R He O Renaudin V-M Kolmonen et al ldquoCharacterizationof quasi-stationarity regions for vehicle-to-vehicle radiochannelsrdquo IEEE Transactions on Antennas and Propagationvol 63 no 5 pp 2237ndash2251 2015

[11] YMa L Yang andX Zheng ldquoA geometry-based non-stationaryMIMO channel model for vehicular communicationsrdquo ChinaCommunications vol 15 no 7 pp 30ndash38 2018

[12] C A Gutirrez J T Gutirrez-Mena J M Luna-RiveraD U Campos-Delgado R Velzquez and M Patzold ldquoGe-ometry-based statistical modeling of non-WSSUS mobile-to-mobile Rayleigh fading channelsrdquo IEEE Transactions onVehicular Technology vol 67 no 1 pp 362ndash377 2018

[13] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the Vehicular Technology Confer-ence pp 1ndash6 San Francisco CA USA September 2011

[14] A Borhani and M Patzold ldquoCorrelation and spectralproperties of vehicle-to-vehicle channels in the presence ofmoving scatterersrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 9 pp 4228ndash4239 2013

[15] H Jiang Z Zhang L Wu and J Dang ldquoNovel 3-D irregular-shaped geometry-based channel modeling for semi-ellipsoidvehicle-to-vehicle scattering environmentsrdquo IEEE WirelessCommunications Letters vol 7 no 5 pp 836ndash839 2018

[16] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014

[17] X Zhao X Liang S Li and B Ai ldquoTwo-cylinder and multi-ring GBSSM for realizing and modeling of vehicle-to-vehiclewideband MIMO channelsrdquo IEEE Transactions on IntelligentTransportation Systems vol 17 no 10 pp 2787ndash2799 2016

[18] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA novel 3Dnonstationary channel model based on the von Mises-Fisherscattering distributionrdquo Mobile Information Systemsvol 2016 Article ID 2161460 pp 1ndash9 2016

[19] Y Yuan C-X Wang Y He M M Alwakeel andE-H M Aggoune ldquo3D wideband non-stationary geometry-based stochastic models for non-isotropic MIMO vehicle-to-vehicle channelsrdquo IEEE Transactions on Wireless Communi-cations vol 14 no 12 pp 6883ndash6895 2015

[20] N Avazov S M R Islam D Park and K S Kwak ldquoStatisticalcharacterization of a 3-D propagation model for V2Vchannels in rectangular tunnelsrdquo IEEE Antennas andWirelessPropagation Letters vol 16 pp 2392ndash2395 2017

[21] H Jiang Z Zhang J Dang and L Wu ldquoA novel 3-D massiveMIMO channel model for vehicle-to-vehicle communicationenvironmentsrdquo IEEE Transactions on Communicationsvol 66 no 1 pp 79ndash90 2018

[22] N Avazov S M R Islam D Park and K S Kwak ldquoCluster-based non-stationary channel modeling for vehicle-to-vehiclecommunicationsrdquo IEEE Antennas and Wireless PropagationLetters vol 16 pp 1419ndash1422 2016

International Journal of Antennas and Propagation 11

[23] D Du X Zeng X Jian L Miao and H Wang ldquoree-di-mensional vehicle-to-vehicle channel modeling with multiplemoving scatterersrdquo Mobile Information Systems vol 2017Article ID 7231417 pp 1ndash14 2017

[24] X LiangW Cao and X Zhao ldquoDoppler power spectra for 3Dvehicle-to-vehicle channels with moving scatterersrdquo IEEEAccess vol 6 pp 42822ndash42828 2018

[25] X Chen Y Fang Y Sun Y Pan and W Xiang ldquoStatisticalcharacterization of novel 3D cluster-based MIMO vehicle-to-vehicle models for urban street scattering environmentsrdquoInternational Journal of Antennas and Propagation vol 2018Article ID 6742346 pp 1ndash11 2018

[26] Q Zhu H Li Y Fu et al ldquoA novel 3D non-stationary wirelessMIMO channel simulator and hardware emulatorrdquo IEEETransactions on Communications vol 66 no 9 pp 3865ndash3878 2018

[27] K Jiang X Chen Q Zhu L Chen D Xu and B Chen ldquoAnovel simulation model for nonstationary rice fading chan-nelsrdquo Wireless Communications and Mobile Computingvol 2018 Article ID 8086073 pp 1ndash9 2018

[28] M Patzold C A Gutierrez and N Youssef ldquoOn the con-sistency of non-stationary multipath fading channels withrespect to the average doppler shift and the doppler spreadrdquoin Proceedings of the IEEE Wireless Communications andNetworking Conference (WCNC) pp 1ndash6 San Francisco CAUSA 2017

[29] Q Zhu W Li C-X Wang et al ldquoTemporal correlations for anon-stationary vehicle-to-vehicle channel model allowingvelocity variationsrdquo IEEE Communications Letters vol 23no 7 pp 1280ndash1284 2019

[30] Q Zhu Y Yang X Chen et al ldquoA novel 3D non-stationaryvehicle-to-vehicle channel model and its spatial-temporalcorrelation propertiesrdquo IEEE Access vol 6 pp 43633ndash436432018

[31] W Dahech M Patzold and N Youssef ldquoA non-stationarymobile-to-mobile multipath fading channel model takingaccount of velocity variations of the mobile stationsrdquo inProceedings of the IEEE International Symposium on Antennasand Propagation pp 1ndash4 Lisbon Portugal April 2015

[32] W Dahech M Patzold C A Gutierrez and N Youssef ldquoAnon-stationary mobile-to-mobile channel model allowing forvelocity and trajectory variations of the mobile stationsrdquo IEEETransactions on Wireless Communications vol 16 no 3pp 1987ndash2000 2017

[33] C A Gutierrez M Patzold W Dahech and N Youssef ldquoAnon-WSSUS mobile-to-mobile channel model assuming ve-locity variations of the mobile stationsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 1ndash6 San Francisco CA USA March 2017

[34] Q Zhu Y Yang C X Wang et al ldquoSpatial correlations of a3D non-stationary MIMO channel model with 3D antennaarrays and 3D arbitrary trajectoriesrdquo IEEE Wireless Com-munications Letters vol 8 no 2 pp 512ndash515 2019

[35] Y Yang Q Zhu W Li et al ldquoA general 3D non-stationarytwin-cluster model for vehicle-to-vehicle MIMO channelsrdquo inProceedings of the Wireless Communications and SignalProcessing pp 1ndash6 Hangzhou China Octobor 2018

[36] H Hofstertter A F Molisch and N Czink ldquoA twin-clusterMIMO channel modelrdquo in Proceedings of the Antennas andPropagation (EuCAP) pp 1ndash8 Nice France November 2006

[37] I Sen and D W Matolak ldquoVehicle-vehicle channel modelsfor the 5-GHz bandrdquo IEEE Transactions on IntelligentTransportation Systems vol 9 no 2 pp 235ndash245 2008

[38] T Zwick C Fischer D Didascalou and W Wiesbeck ldquoAstochastic spatial channel model based on wave-propagationmodelingrdquo IEEE Journal on Selected Areas in Communica-tions vol 18 no 1 pp 6ndash15 2000

[39] K Mammasis and RW Stewart ldquoe Fisher-Bingham spatialcorrelation model for multielement antenna systemsrdquo IEEETransactions on Vehicular Technology vol 58 no 5pp 2130ndash2136 2009

[40] M Konstantinos S W Robert and J S ompson ldquoSpatialfading correlation model using mixtures of Von Mises Fisherdistributionsrdquo IEEE Transactions on Wireless Communica-tions vol 8 no 4 pp 2046ndash2055 2009

[41] A Fayziyev M Patzold E Masson Y Cocheril andM Berbineau ldquoA measurement-based channel model forvehicular communications in tunnelsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 116ndash121 Istanbul Turkey April 2014

[42] S Payami and F Tufvesson ldquoChannel measurements andanalysis for very large array systems at 26 GHzrdquo in Pro-ceedings of the Antennas and Propagation (EUCAP)pp 433ndash437 Prague Czech Republic March 2012

12 International Journal of Antennas and Propagation

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

which is similar at two terminals in the communicationsystem By substituting u1 s1 1 u2 s2 2 and n 1into (21) and (33) we can get the theoretical and approx-imated values of SCFs In addition the measured results in[42] are also shown in Figure 6(a) e good agreement ofSCFs shows the correctness of both the theoretical modeland derivations Similarly by substituting u s 1 and n

1 into (36) and (43) we obtain and show the theoretical andapproximated values of TCFs in Figure 6(b) e goodagreements between theoretical simulated and approxi-mated results reveal that our simulation and derivation arecorrect

Finally substituting TCFs into (44) we can get thetheoretical and simulated values of DPSDs for Path IIIwhich are also shown in Figure 7 It clearly shows thatmoving clusters and two terminals have an influence on thedrifting of DPSDs Moreover good agreements between thesimulated and theoretical results indicate that our proposedmodel is correct

6 Conclusion

is paper has proposed a general channel model charac-terized by 3D scattering environments 3D movements and

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)Measurement

0 05 1 15 2

t = 4s

t = 8s

t = 0s

Normalized antenna space (Δdλ)

Abso

lute

val

ue o

f SCF

(a)

0

02

04

06

08

1

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

0 0005 001 0015 002 0025 003 0035 004

Abso

lute

val

ue o

f TCF

Time lag (Δt)

t = 8s

(b)

Figure 6 Comparison of absolute theoretical and simulated (a) SCFs and (b) TCFs at different time instants

86

4ndash5

0

ndash1000

5

2

10

ndash500

15

0 500 01000Frequency f (HZ)

DPS

D

Time t (s)

(a)

0

5

10

15

Path III (theoretical results)Path III (simulated results)

t = 4s

t = 0s

t = 8s

ndash600 ndash400 ndash200 0 200 400 600Frequency f (HZ)

DPS

D

(b)

Figure 7 (a) eoretical 3D DPSD and (b) theoretical and simulated DPSDs at different time instants for Path III

10 International Journal of Antennas and Propagation

3D-shaped antenna arrays of two terminals for V2V com-munication system e upgraded algorithms for time-evolving parameters such as the number of paths pathdelays path powers and angles have been developed andanalyzed in detail Meanwhile the approximated expressionsof statistical properties including SCF TCF and DPSD havebeen derived and verified by simulations Simulated andanalyzed results show that different moving trajectoriessignificantly affect the V2V channel characteristic Mean-while the generalization and compatibility of our proposedmodel also demonstrate that some previous models with 2Dor even 1D movements have special applications e newmodel can be used to develop analyze and test realistic V2Vcommunication systems in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Key ScientificInstrument and Equipment Development Project (Grant no61827801) Aeronautical Science Foundation of China(Grant no 2017ZC52021) and Open Foundation forGraduate Innovation of NUAA (Grant no KFJJ 20180408)

References

[1] C-X Wang X Cheng and D I Laurenson ldquoVehicle-to-vehicle channel modeling andmeasurements recent advancesand future challengesrdquo IEEE Communications Magazinevol 47 no 11 pp 96ndash103 2009

[2] K Guan B Ai M Liso Nicolas et al ldquoOn the influence ofscattering from traffic signs in vehicle-to-x communicationsrdquoIEEE Transactions on Vehicular Technology vol 65 no 8pp 5835ndash5849 2016

[3] X Cheng C-X Wang B Ai and H Aggoune ldquoEnvelopelevel crossing rate and average fade duration of nonisotropicvehicle-to-vehicle ricean fading channelsrdquo IEEE Transactionson Intelligent Transportation Systems vol 15 no 1 pp 62ndash722014

[4] Q Zhu C Xue X Chen and Y Yang ldquoA newMIMO channelmodel incorporating antenna effectsrdquo Progress In Electro-magnetics Research M vol 50 pp 129ndash140 2016

[5] A G Zajic and G L Stubber ldquoSpace-time correlated mobile-to-mobile channels modelling and simulationrdquo IEEETransactions on Vehicular Technology vol 57 no 2pp 715ndash726 2008

[6] W Fan X Carreno Bautista de Lisbona F Sun J O NielsenM B Knudsen and G F Pedersen ldquoEmulating spatialcharacteristics of MIMO channels for OTA testingrdquo IEEETransactions on Antennas and Propagation vol 61 no 8pp 4306ndash4314 2013

[7] J Zhang Y Zhang Y Yu R Xu Q Zheng and P Zhangldquo3-D MIMO how much does it meet our expectationsobserved from channel measurementsrdquo IEEE Journal on

Selected Areas in Communications vol 35 no 8pp 1887ndash1903 2017

[8] J Karedal F Tufvesson N Czink et al ldquoA geometry-basedstochastic MIMO model for vehicle-to-vehicle communica-tionsrdquo IEEE Transactions onWireless Communications vol 8no 7 pp 3646ndash3657 2009

[9] A Paier T Zemen L Bernado et al ldquoNon-WSSUS vehicularchannel characterization in high-way and urban scenarios at52 GHz using the local scattering functionrdquo in Proceedings ofthe 2008 International ITG Workshop on Smart Antennaspp 9ndash15 Darmstadt Germany February 2008

[10] R He O Renaudin V-M Kolmonen et al ldquoCharacterizationof quasi-stationarity regions for vehicle-to-vehicle radiochannelsrdquo IEEE Transactions on Antennas and Propagationvol 63 no 5 pp 2237ndash2251 2015

[11] YMa L Yang andX Zheng ldquoA geometry-based non-stationaryMIMO channel model for vehicular communicationsrdquo ChinaCommunications vol 15 no 7 pp 30ndash38 2018

[12] C A Gutirrez J T Gutirrez-Mena J M Luna-RiveraD U Campos-Delgado R Velzquez and M Patzold ldquoGe-ometry-based statistical modeling of non-WSSUS mobile-to-mobile Rayleigh fading channelsrdquo IEEE Transactions onVehicular Technology vol 67 no 1 pp 362ndash377 2018

[13] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the Vehicular Technology Confer-ence pp 1ndash6 San Francisco CA USA September 2011

[14] A Borhani and M Patzold ldquoCorrelation and spectralproperties of vehicle-to-vehicle channels in the presence ofmoving scatterersrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 9 pp 4228ndash4239 2013

[15] H Jiang Z Zhang L Wu and J Dang ldquoNovel 3-D irregular-shaped geometry-based channel modeling for semi-ellipsoidvehicle-to-vehicle scattering environmentsrdquo IEEE WirelessCommunications Letters vol 7 no 5 pp 836ndash839 2018

[16] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014

[17] X Zhao X Liang S Li and B Ai ldquoTwo-cylinder and multi-ring GBSSM for realizing and modeling of vehicle-to-vehiclewideband MIMO channelsrdquo IEEE Transactions on IntelligentTransportation Systems vol 17 no 10 pp 2787ndash2799 2016

[18] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA novel 3Dnonstationary channel model based on the von Mises-Fisherscattering distributionrdquo Mobile Information Systemsvol 2016 Article ID 2161460 pp 1ndash9 2016

[19] Y Yuan C-X Wang Y He M M Alwakeel andE-H M Aggoune ldquo3D wideband non-stationary geometry-based stochastic models for non-isotropic MIMO vehicle-to-vehicle channelsrdquo IEEE Transactions on Wireless Communi-cations vol 14 no 12 pp 6883ndash6895 2015

[20] N Avazov S M R Islam D Park and K S Kwak ldquoStatisticalcharacterization of a 3-D propagation model for V2Vchannels in rectangular tunnelsrdquo IEEE Antennas andWirelessPropagation Letters vol 16 pp 2392ndash2395 2017

[21] H Jiang Z Zhang J Dang and L Wu ldquoA novel 3-D massiveMIMO channel model for vehicle-to-vehicle communicationenvironmentsrdquo IEEE Transactions on Communicationsvol 66 no 1 pp 79ndash90 2018

[22] N Avazov S M R Islam D Park and K S Kwak ldquoCluster-based non-stationary channel modeling for vehicle-to-vehiclecommunicationsrdquo IEEE Antennas and Wireless PropagationLetters vol 16 pp 1419ndash1422 2016

International Journal of Antennas and Propagation 11

[23] D Du X Zeng X Jian L Miao and H Wang ldquoree-di-mensional vehicle-to-vehicle channel modeling with multiplemoving scatterersrdquo Mobile Information Systems vol 2017Article ID 7231417 pp 1ndash14 2017

[24] X LiangW Cao and X Zhao ldquoDoppler power spectra for 3Dvehicle-to-vehicle channels with moving scatterersrdquo IEEEAccess vol 6 pp 42822ndash42828 2018

[25] X Chen Y Fang Y Sun Y Pan and W Xiang ldquoStatisticalcharacterization of novel 3D cluster-based MIMO vehicle-to-vehicle models for urban street scattering environmentsrdquoInternational Journal of Antennas and Propagation vol 2018Article ID 6742346 pp 1ndash11 2018

[26] Q Zhu H Li Y Fu et al ldquoA novel 3D non-stationary wirelessMIMO channel simulator and hardware emulatorrdquo IEEETransactions on Communications vol 66 no 9 pp 3865ndash3878 2018

[27] K Jiang X Chen Q Zhu L Chen D Xu and B Chen ldquoAnovel simulation model for nonstationary rice fading chan-nelsrdquo Wireless Communications and Mobile Computingvol 2018 Article ID 8086073 pp 1ndash9 2018

[28] M Patzold C A Gutierrez and N Youssef ldquoOn the con-sistency of non-stationary multipath fading channels withrespect to the average doppler shift and the doppler spreadrdquoin Proceedings of the IEEE Wireless Communications andNetworking Conference (WCNC) pp 1ndash6 San Francisco CAUSA 2017

[29] Q Zhu W Li C-X Wang et al ldquoTemporal correlations for anon-stationary vehicle-to-vehicle channel model allowingvelocity variationsrdquo IEEE Communications Letters vol 23no 7 pp 1280ndash1284 2019

[30] Q Zhu Y Yang X Chen et al ldquoA novel 3D non-stationaryvehicle-to-vehicle channel model and its spatial-temporalcorrelation propertiesrdquo IEEE Access vol 6 pp 43633ndash436432018

[31] W Dahech M Patzold and N Youssef ldquoA non-stationarymobile-to-mobile multipath fading channel model takingaccount of velocity variations of the mobile stationsrdquo inProceedings of the IEEE International Symposium on Antennasand Propagation pp 1ndash4 Lisbon Portugal April 2015

[32] W Dahech M Patzold C A Gutierrez and N Youssef ldquoAnon-stationary mobile-to-mobile channel model allowing forvelocity and trajectory variations of the mobile stationsrdquo IEEETransactions on Wireless Communications vol 16 no 3pp 1987ndash2000 2017

[33] C A Gutierrez M Patzold W Dahech and N Youssef ldquoAnon-WSSUS mobile-to-mobile channel model assuming ve-locity variations of the mobile stationsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 1ndash6 San Francisco CA USA March 2017

[34] Q Zhu Y Yang C X Wang et al ldquoSpatial correlations of a3D non-stationary MIMO channel model with 3D antennaarrays and 3D arbitrary trajectoriesrdquo IEEE Wireless Com-munications Letters vol 8 no 2 pp 512ndash515 2019

[35] Y Yang Q Zhu W Li et al ldquoA general 3D non-stationarytwin-cluster model for vehicle-to-vehicle MIMO channelsrdquo inProceedings of the Wireless Communications and SignalProcessing pp 1ndash6 Hangzhou China Octobor 2018

[36] H Hofstertter A F Molisch and N Czink ldquoA twin-clusterMIMO channel modelrdquo in Proceedings of the Antennas andPropagation (EuCAP) pp 1ndash8 Nice France November 2006

[37] I Sen and D W Matolak ldquoVehicle-vehicle channel modelsfor the 5-GHz bandrdquo IEEE Transactions on IntelligentTransportation Systems vol 9 no 2 pp 235ndash245 2008

[38] T Zwick C Fischer D Didascalou and W Wiesbeck ldquoAstochastic spatial channel model based on wave-propagationmodelingrdquo IEEE Journal on Selected Areas in Communica-tions vol 18 no 1 pp 6ndash15 2000

[39] K Mammasis and RW Stewart ldquoe Fisher-Bingham spatialcorrelation model for multielement antenna systemsrdquo IEEETransactions on Vehicular Technology vol 58 no 5pp 2130ndash2136 2009

[40] M Konstantinos S W Robert and J S ompson ldquoSpatialfading correlation model using mixtures of Von Mises Fisherdistributionsrdquo IEEE Transactions on Wireless Communica-tions vol 8 no 4 pp 2046ndash2055 2009

[41] A Fayziyev M Patzold E Masson Y Cocheril andM Berbineau ldquoA measurement-based channel model forvehicular communications in tunnelsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 116ndash121 Istanbul Turkey April 2014

[42] S Payami and F Tufvesson ldquoChannel measurements andanalysis for very large array systems at 26 GHzrdquo in Pro-ceedings of the Antennas and Propagation (EUCAP)pp 433ndash437 Prague Czech Republic March 2012

12 International Journal of Antennas and Propagation

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

3D-shaped antenna arrays of two terminals for V2V com-munication system e upgraded algorithms for time-evolving parameters such as the number of paths pathdelays path powers and angles have been developed andanalyzed in detail Meanwhile the approximated expressionsof statistical properties including SCF TCF and DPSD havebeen derived and verified by simulations Simulated andanalyzed results show that different moving trajectoriessignificantly affect the V2V channel characteristic Mean-while the generalization and compatibility of our proposedmodel also demonstrate that some previous models with 2Dor even 1D movements have special applications e newmodel can be used to develop analyze and test realistic V2Vcommunication systems in the future

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the National Key ScientificInstrument and Equipment Development Project (Grant no61827801) Aeronautical Science Foundation of China(Grant no 2017ZC52021) and Open Foundation forGraduate Innovation of NUAA (Grant no KFJJ 20180408)

References

[1] C-X Wang X Cheng and D I Laurenson ldquoVehicle-to-vehicle channel modeling andmeasurements recent advancesand future challengesrdquo IEEE Communications Magazinevol 47 no 11 pp 96ndash103 2009

[2] K Guan B Ai M Liso Nicolas et al ldquoOn the influence ofscattering from traffic signs in vehicle-to-x communicationsrdquoIEEE Transactions on Vehicular Technology vol 65 no 8pp 5835ndash5849 2016

[3] X Cheng C-X Wang B Ai and H Aggoune ldquoEnvelopelevel crossing rate and average fade duration of nonisotropicvehicle-to-vehicle ricean fading channelsrdquo IEEE Transactionson Intelligent Transportation Systems vol 15 no 1 pp 62ndash722014

[4] Q Zhu C Xue X Chen and Y Yang ldquoA newMIMO channelmodel incorporating antenna effectsrdquo Progress In Electro-magnetics Research M vol 50 pp 129ndash140 2016

[5] A G Zajic and G L Stubber ldquoSpace-time correlated mobile-to-mobile channels modelling and simulationrdquo IEEETransactions on Vehicular Technology vol 57 no 2pp 715ndash726 2008

[6] W Fan X Carreno Bautista de Lisbona F Sun J O NielsenM B Knudsen and G F Pedersen ldquoEmulating spatialcharacteristics of MIMO channels for OTA testingrdquo IEEETransactions on Antennas and Propagation vol 61 no 8pp 4306ndash4314 2013

[7] J Zhang Y Zhang Y Yu R Xu Q Zheng and P Zhangldquo3-D MIMO how much does it meet our expectationsobserved from channel measurementsrdquo IEEE Journal on

Selected Areas in Communications vol 35 no 8pp 1887ndash1903 2017

[8] J Karedal F Tufvesson N Czink et al ldquoA geometry-basedstochastic MIMO model for vehicle-to-vehicle communica-tionsrdquo IEEE Transactions onWireless Communications vol 8no 7 pp 3646ndash3657 2009

[9] A Paier T Zemen L Bernado et al ldquoNon-WSSUS vehicularchannel characterization in high-way and urban scenarios at52 GHz using the local scattering functionrdquo in Proceedings ofthe 2008 International ITG Workshop on Smart Antennaspp 9ndash15 Darmstadt Germany February 2008

[10] R He O Renaudin V-M Kolmonen et al ldquoCharacterizationof quasi-stationarity regions for vehicle-to-vehicle radiochannelsrdquo IEEE Transactions on Antennas and Propagationvol 63 no 5 pp 2237ndash2251 2015

[11] YMa L Yang andX Zheng ldquoA geometry-based non-stationaryMIMO channel model for vehicular communicationsrdquo ChinaCommunications vol 15 no 7 pp 30ndash38 2018

[12] C A Gutirrez J T Gutirrez-Mena J M Luna-RiveraD U Campos-Delgado R Velzquez and M Patzold ldquoGe-ometry-based statistical modeling of non-WSSUS mobile-to-mobile Rayleigh fading channelsrdquo IEEE Transactions onVehicular Technology vol 67 no 1 pp 362ndash377 2018

[13] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the Vehicular Technology Confer-ence pp 1ndash6 San Francisco CA USA September 2011

[14] A Borhani and M Patzold ldquoCorrelation and spectralproperties of vehicle-to-vehicle channels in the presence ofmoving scatterersrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 9 pp 4228ndash4239 2013

[15] H Jiang Z Zhang L Wu and J Dang ldquoNovel 3-D irregular-shaped geometry-based channel modeling for semi-ellipsoidvehicle-to-vehicle scattering environmentsrdquo IEEE WirelessCommunications Letters vol 7 no 5 pp 836ndash839 2018

[16] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014

[17] X Zhao X Liang S Li and B Ai ldquoTwo-cylinder and multi-ring GBSSM for realizing and modeling of vehicle-to-vehiclewideband MIMO channelsrdquo IEEE Transactions on IntelligentTransportation Systems vol 17 no 10 pp 2787ndash2799 2016

[18] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA novel 3Dnonstationary channel model based on the von Mises-Fisherscattering distributionrdquo Mobile Information Systemsvol 2016 Article ID 2161460 pp 1ndash9 2016

[19] Y Yuan C-X Wang Y He M M Alwakeel andE-H M Aggoune ldquo3D wideband non-stationary geometry-based stochastic models for non-isotropic MIMO vehicle-to-vehicle channelsrdquo IEEE Transactions on Wireless Communi-cations vol 14 no 12 pp 6883ndash6895 2015

[20] N Avazov S M R Islam D Park and K S Kwak ldquoStatisticalcharacterization of a 3-D propagation model for V2Vchannels in rectangular tunnelsrdquo IEEE Antennas andWirelessPropagation Letters vol 16 pp 2392ndash2395 2017

[21] H Jiang Z Zhang J Dang and L Wu ldquoA novel 3-D massiveMIMO channel model for vehicle-to-vehicle communicationenvironmentsrdquo IEEE Transactions on Communicationsvol 66 no 1 pp 79ndash90 2018

[22] N Avazov S M R Islam D Park and K S Kwak ldquoCluster-based non-stationary channel modeling for vehicle-to-vehiclecommunicationsrdquo IEEE Antennas and Wireless PropagationLetters vol 16 pp 1419ndash1422 2016

International Journal of Antennas and Propagation 11

[23] D Du X Zeng X Jian L Miao and H Wang ldquoree-di-mensional vehicle-to-vehicle channel modeling with multiplemoving scatterersrdquo Mobile Information Systems vol 2017Article ID 7231417 pp 1ndash14 2017

[24] X LiangW Cao and X Zhao ldquoDoppler power spectra for 3Dvehicle-to-vehicle channels with moving scatterersrdquo IEEEAccess vol 6 pp 42822ndash42828 2018

[25] X Chen Y Fang Y Sun Y Pan and W Xiang ldquoStatisticalcharacterization of novel 3D cluster-based MIMO vehicle-to-vehicle models for urban street scattering environmentsrdquoInternational Journal of Antennas and Propagation vol 2018Article ID 6742346 pp 1ndash11 2018

[26] Q Zhu H Li Y Fu et al ldquoA novel 3D non-stationary wirelessMIMO channel simulator and hardware emulatorrdquo IEEETransactions on Communications vol 66 no 9 pp 3865ndash3878 2018

[27] K Jiang X Chen Q Zhu L Chen D Xu and B Chen ldquoAnovel simulation model for nonstationary rice fading chan-nelsrdquo Wireless Communications and Mobile Computingvol 2018 Article ID 8086073 pp 1ndash9 2018

[28] M Patzold C A Gutierrez and N Youssef ldquoOn the con-sistency of non-stationary multipath fading channels withrespect to the average doppler shift and the doppler spreadrdquoin Proceedings of the IEEE Wireless Communications andNetworking Conference (WCNC) pp 1ndash6 San Francisco CAUSA 2017

[29] Q Zhu W Li C-X Wang et al ldquoTemporal correlations for anon-stationary vehicle-to-vehicle channel model allowingvelocity variationsrdquo IEEE Communications Letters vol 23no 7 pp 1280ndash1284 2019

[30] Q Zhu Y Yang X Chen et al ldquoA novel 3D non-stationaryvehicle-to-vehicle channel model and its spatial-temporalcorrelation propertiesrdquo IEEE Access vol 6 pp 43633ndash436432018

[31] W Dahech M Patzold and N Youssef ldquoA non-stationarymobile-to-mobile multipath fading channel model takingaccount of velocity variations of the mobile stationsrdquo inProceedings of the IEEE International Symposium on Antennasand Propagation pp 1ndash4 Lisbon Portugal April 2015

[32] W Dahech M Patzold C A Gutierrez and N Youssef ldquoAnon-stationary mobile-to-mobile channel model allowing forvelocity and trajectory variations of the mobile stationsrdquo IEEETransactions on Wireless Communications vol 16 no 3pp 1987ndash2000 2017

[33] C A Gutierrez M Patzold W Dahech and N Youssef ldquoAnon-WSSUS mobile-to-mobile channel model assuming ve-locity variations of the mobile stationsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 1ndash6 San Francisco CA USA March 2017

[34] Q Zhu Y Yang C X Wang et al ldquoSpatial correlations of a3D non-stationary MIMO channel model with 3D antennaarrays and 3D arbitrary trajectoriesrdquo IEEE Wireless Com-munications Letters vol 8 no 2 pp 512ndash515 2019

[35] Y Yang Q Zhu W Li et al ldquoA general 3D non-stationarytwin-cluster model for vehicle-to-vehicle MIMO channelsrdquo inProceedings of the Wireless Communications and SignalProcessing pp 1ndash6 Hangzhou China Octobor 2018

[36] H Hofstertter A F Molisch and N Czink ldquoA twin-clusterMIMO channel modelrdquo in Proceedings of the Antennas andPropagation (EuCAP) pp 1ndash8 Nice France November 2006

[37] I Sen and D W Matolak ldquoVehicle-vehicle channel modelsfor the 5-GHz bandrdquo IEEE Transactions on IntelligentTransportation Systems vol 9 no 2 pp 235ndash245 2008

[38] T Zwick C Fischer D Didascalou and W Wiesbeck ldquoAstochastic spatial channel model based on wave-propagationmodelingrdquo IEEE Journal on Selected Areas in Communica-tions vol 18 no 1 pp 6ndash15 2000

[39] K Mammasis and RW Stewart ldquoe Fisher-Bingham spatialcorrelation model for multielement antenna systemsrdquo IEEETransactions on Vehicular Technology vol 58 no 5pp 2130ndash2136 2009

[40] M Konstantinos S W Robert and J S ompson ldquoSpatialfading correlation model using mixtures of Von Mises Fisherdistributionsrdquo IEEE Transactions on Wireless Communica-tions vol 8 no 4 pp 2046ndash2055 2009

[41] A Fayziyev M Patzold E Masson Y Cocheril andM Berbineau ldquoA measurement-based channel model forvehicular communications in tunnelsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 116ndash121 Istanbul Turkey April 2014

[42] S Payami and F Tufvesson ldquoChannel measurements andanalysis for very large array systems at 26 GHzrdquo in Pro-ceedings of the Antennas and Propagation (EUCAP)pp 433ndash437 Prague Czech Republic March 2012

12 International Journal of Antennas and Propagation

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

[23] D Du X Zeng X Jian L Miao and H Wang ldquoree-di-mensional vehicle-to-vehicle channel modeling with multiplemoving scatterersrdquo Mobile Information Systems vol 2017Article ID 7231417 pp 1ndash14 2017

[24] X LiangW Cao and X Zhao ldquoDoppler power spectra for 3Dvehicle-to-vehicle channels with moving scatterersrdquo IEEEAccess vol 6 pp 42822ndash42828 2018

[25] X Chen Y Fang Y Sun Y Pan and W Xiang ldquoStatisticalcharacterization of novel 3D cluster-based MIMO vehicle-to-vehicle models for urban street scattering environmentsrdquoInternational Journal of Antennas and Propagation vol 2018Article ID 6742346 pp 1ndash11 2018

[26] Q Zhu H Li Y Fu et al ldquoA novel 3D non-stationary wirelessMIMO channel simulator and hardware emulatorrdquo IEEETransactions on Communications vol 66 no 9 pp 3865ndash3878 2018

[27] K Jiang X Chen Q Zhu L Chen D Xu and B Chen ldquoAnovel simulation model for nonstationary rice fading chan-nelsrdquo Wireless Communications and Mobile Computingvol 2018 Article ID 8086073 pp 1ndash9 2018

[28] M Patzold C A Gutierrez and N Youssef ldquoOn the con-sistency of non-stationary multipath fading channels withrespect to the average doppler shift and the doppler spreadrdquoin Proceedings of the IEEE Wireless Communications andNetworking Conference (WCNC) pp 1ndash6 San Francisco CAUSA 2017

[29] Q Zhu W Li C-X Wang et al ldquoTemporal correlations for anon-stationary vehicle-to-vehicle channel model allowingvelocity variationsrdquo IEEE Communications Letters vol 23no 7 pp 1280ndash1284 2019

[30] Q Zhu Y Yang X Chen et al ldquoA novel 3D non-stationaryvehicle-to-vehicle channel model and its spatial-temporalcorrelation propertiesrdquo IEEE Access vol 6 pp 43633ndash436432018

[31] W Dahech M Patzold and N Youssef ldquoA non-stationarymobile-to-mobile multipath fading channel model takingaccount of velocity variations of the mobile stationsrdquo inProceedings of the IEEE International Symposium on Antennasand Propagation pp 1ndash4 Lisbon Portugal April 2015

[32] W Dahech M Patzold C A Gutierrez and N Youssef ldquoAnon-stationary mobile-to-mobile channel model allowing forvelocity and trajectory variations of the mobile stationsrdquo IEEETransactions on Wireless Communications vol 16 no 3pp 1987ndash2000 2017

[33] C A Gutierrez M Patzold W Dahech and N Youssef ldquoAnon-WSSUS mobile-to-mobile channel model assuming ve-locity variations of the mobile stationsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 1ndash6 San Francisco CA USA March 2017

[34] Q Zhu Y Yang C X Wang et al ldquoSpatial correlations of a3D non-stationary MIMO channel model with 3D antennaarrays and 3D arbitrary trajectoriesrdquo IEEE Wireless Com-munications Letters vol 8 no 2 pp 512ndash515 2019

[35] Y Yang Q Zhu W Li et al ldquoA general 3D non-stationarytwin-cluster model for vehicle-to-vehicle MIMO channelsrdquo inProceedings of the Wireless Communications and SignalProcessing pp 1ndash6 Hangzhou China Octobor 2018

[36] H Hofstertter A F Molisch and N Czink ldquoA twin-clusterMIMO channel modelrdquo in Proceedings of the Antennas andPropagation (EuCAP) pp 1ndash8 Nice France November 2006

[37] I Sen and D W Matolak ldquoVehicle-vehicle channel modelsfor the 5-GHz bandrdquo IEEE Transactions on IntelligentTransportation Systems vol 9 no 2 pp 235ndash245 2008

[38] T Zwick C Fischer D Didascalou and W Wiesbeck ldquoAstochastic spatial channel model based on wave-propagationmodelingrdquo IEEE Journal on Selected Areas in Communica-tions vol 18 no 1 pp 6ndash15 2000

[39] K Mammasis and RW Stewart ldquoe Fisher-Bingham spatialcorrelation model for multielement antenna systemsrdquo IEEETransactions on Vehicular Technology vol 58 no 5pp 2130ndash2136 2009

[40] M Konstantinos S W Robert and J S ompson ldquoSpatialfading correlation model using mixtures of Von Mises Fisherdistributionsrdquo IEEE Transactions on Wireless Communica-tions vol 8 no 4 pp 2046ndash2055 2009

[41] A Fayziyev M Patzold E Masson Y Cocheril andM Berbineau ldquoA measurement-based channel model forvehicular communications in tunnelsrdquo in Proceedings of theWireless Communications and Networking Conference(WCNC) pp 116ndash121 Istanbul Turkey April 2014

[42] S Payami and F Tufvesson ldquoChannel measurements andanalysis for very large array systems at 26 GHzrdquo in Pro-ceedings of the Antennas and Propagation (EUCAP)pp 433ndash437 Prague Czech Republic March 2012

12 International Journal of Antennas and Propagation

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

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