A 3D Non-Stationary Wideband Geometry-Based Channel Model for MIMO Vehicle-to-Vehicle Communication System

1 A 3D Non-Stationary W ideband Geometry-Based Channel Model for MIMO V ehicle- to-V ehicl e Communication Sy stems Hao Jiang, Zaichen Zhang, Senior Member , IEEE , Lian g W u, Membe r , I EEE , Jian Dang, Member , IEEE , and Guan Gui, Senior Mem ber , IEEE Abstract —In this paper , we present a thr ee-dimensional (3D) non-wide-sense stationary (non-WSS) wideband geom etry-based channel model fo r vehicle-to-v ehicle (V2V) communication en- vironments. W e in troduce a two-cylinder model to describe moving vehicles as well as multiple confo cal semi-ellipsoid models to d epict stationary roadside scenarios. Th e received signal is constructed as a sum of the line-of-sight (LoS), single-, and double-boun ced rays with different energies. Accordin gly , the proposed channel model is su fﬁcient for depi cting a wide variety of V2V env ironments, such as macro-, micro-, and picocells. The relati ve movement between t he mobile transmitter (MT) and mobile r eceiv er (MR) results in time-varia nt geometr ic statistics that make our chann el model non-stationary . Usin g this chann el model, the p roposed channel statistics, i.e., th e time-variant space correla tion functions (CF s), frequency CFs, and corresponding Doppler power sp ectral density (PS D), were studied for different relati ve mo ving time instants. The nu merical results demonstrate that the p roposed 3D non-WSS wideband chann el model is practical f or characterizing r eal V2V channels. Index T erms —3D channel model, vehicle-to-vehicle communi- cation en vironments, time-variant space and frequency corr ela- tion functions, Doppler power sp ectral density . I . I N T R O D U C T I O N A. Motivation Recently , vehicle-to -vehicle (V2 V) comm unications have received widespread app lica tio ns on accoun t of th e rapid development o f ﬁfth -genera tion (5G) wir eless communicatio n networks [1]. Unlike conv entional ﬁxed-to-m obile (F2 M) cel- lular systems, V2V systems a r e employed with low-ele vation multiple antenn as, and the mob ile transmitter (MT) and mobile receiver (MR) are both in relative motion. In V2V scen arios, multiple-inp ut and multiple-ou tput (MI MO) techn o logy is becoming incre asingly attractive becau se large-scale antenna elements can be easily m ounted on vehicular surfaces. For facilitating the design and an alysis of V2V co mmunicatio n systems, th e rad io p ropag a tion ch aracteristics must be de- signed betwe e n the MT a nd M R [2-4]. Reliable knowledge This work is supported by national key research and de vel opment plan (No. 2016YFB050220 2), NSFC project s (6157110 5, 6150110 9, 61601119 , and 61601120), and scienti ﬁc research foundation of graduate school of Southea st Uni ve rsity (No. YBJJ1655). H. Jiang, Z . Zhang, L. W u, and J. Dang are with the National Mobile Communicat ions Research Laboratory , Southeast Uni versi ty , Nanjing 210096, P . R. China. (e-mails: { jianghao, zczhang, wuliang, newwan da } @seu.edu .cn), Correspondi ng author: zczhang@seu.edu .cn). G. Gui is with the Department of T elecommuni catio n and Information Engineeri ng, Nanjing Uni versit y of Posts and T elecommunica tions, Nanjing 210003, P . R. China (email: guiguan@n jupt.edu.cn ). of th e realistic propa gation channel models, wh ich provide effecti ve and simple mean s to a p proxim ately express the statistical pro perties of the V2V cha n nel [5-6]. B. Prior W ork 1) Geometry-Ba sed Channels T o e valuate the perfor mance of MIMO V2V comm u nication systems, accurate channel m odels are in dispensable. Regardin g the appr o ach of V2V channel modeling , the m o dels c a n be categorized as deter ministic models (mainly in dicates the r ay- tracing metho d) an d stocha stic models. In particular, stochastic models can be ro u ghly divided into several categories as non- geometrica l geom etry-based stoch astic m odels (NGBSMs) and regu la r-shaped geom etry-based stochastic models (RS- GBSMs) [9] -[17]. The fo rmer is also kn own as parametric models, which is constructed based on the ch annel measure- ments, while th e latter is based on the r egu lar g eometric shape of scatterers. In [9], the auth ors dem onstrated that the line-o f-sight (LoS) is more likely to be ob structed by buildings and obstacles between the MT a n d MR. T hus, it is necessary to develop Rayleigh chan nels to describe the V2V environments. The authors of [1 1] introdu ced an RS-GBSM for V2 V scenarios. The auth or presented a two-ring model to dep ict the mov- ing scatterers and multiple con focal ellipses to mimic static scatterers. Y uan [ 1 2] adopted a two-spher e model to describe moving vehicles as well as multiple confocal elliptic-cylinder models to d epict stationary roadside scenarios. In 2014, Z ajic [13] pr oposed a two-cylinder model to dep ict moving and stationary scatterers in the vicin ity of the transmitter and receiver . Accord ingly , in [13 ], the autho r s stated tha t the mobility of scatterers sign iﬁcantly affects th e Dop pler spec- trum; therefo r e, it is im p ortant to accurately acco u nt for tha t effect. Fur th ermore, three-d imensional (3 D) RS-GBSMs for macrocell an d m icrocell commun ication e nvironmen ts wer e respectively presen ted in [1 5] and [1 6 ]. Howe ver , mo st of the above RS-GBSMs focus on nar rowband channel mod els, wherein all ray s experien ce a similar prop agation delay [ 17]. This unrealistically d escribes wire less commu n ication envi- ronmen ts. Accor d ing to the channel measure m ents be twe e n the narrowband an d w id eband V2V cha nnels in [18], Sen conclud e d that the chann el statistics for different time de la y s in wideband channels should be addr e ssed . In 20 09, the author s in [19] ﬁrst pr oposed a w id eband RS-GBSM for MIMO 2 V2V Ricean fading c h annels. Howev er , in [19], th e model was shown to be unable to describe th e chann el statistics for different time delays, which are signiﬁcant for wid eband channels. Based on two m easured scena rios in [20], Cheng [21] introdu ced th e c oncept of high vehicle trafﬁc den sity (VTD) and low VTD to represen t m oving vehicles, respe c- ti vely . That au thor presented the chann el statistics for different time de la y s (i.e., per-tap cha n nel statistics); nevertheless, th e angular spreading of in cident wav es in an elev atio n p la n e in 3D space was igno red. 2) Non-Sta tionary Channels Most pr evious channel mod e ls rely on the wide-sense sta- tionary unco rrelated scattering (WSSUS) a ssumption, which adopts a static channel with co nstant mo del parameters, can not be used to depict the d y namic c h annel prop erties. T o ﬁll the above gaps, it is thus desirable to re-e valuate the v alid ity of non-wide-sense stationar y (non-WSS) chann el modelin g to describe the vehicular comm unications between an M T and MR. In [2 3] and [24], the au thors pr oposed two-dimensiona l (2D) geometry- based no n-WSS nar rowband chan nel models for T -junction and straight r oad en vironmen ts, respectively . Additionally , [ 2 5] and [2 6 ] p resented 2D no n-stationary th eo- retical wideban d MIMO Ricean chann els fo r V 2 V scenarios. Howe ver, these chann els rem ain r estricted to re search in an azim u th plan e. Fur th ermore, Y u a n [27] presented a 3 D wideband MI MO V2V channel. Noneth e less, that auth o r only focused on two r elativ e special movin g dir ections: the same direction and oppo site dir ection. The authors of [28] presen ted a wideban d MIMO model for V2 V chann els based on exten- si ve measur ements taken in high way and r ural environments. In their study , the effects of the mobile d iscr ete scatterers, static d iscrete scatterer s, and diffuse scatterers on the time- variant chan nel p roperties were inv estigated. For the ab ove- mentioned c hannel m odels, they did no t ana ly ze the mobile proper ties between the M T and MR, inclu d ing the relative moving time and m oving directions [5 ]. Howev er , the time- variant space-time and freq uency CFs, which are meaningf ul for th e wir eless chan nel, were not studied in detail. Th erefore, these models canno t realistically descr ib e V2V commun ication en vironmen ts. C. Main Contributions In th is paper, we present a 3D non-station ary wideb and semi-ellipsoid mo del for MIMO V2V Ricean fading channels. The model is operated a t 5.4 GHz, with a bandwidth o f 50 MHz . 1 Compared with the work in [28], th e chan nel model in th is paper is ca p able of depicting a wide variety of 1 Actuall y , the proposed band is also capable of some other V2V en vi- ronments, such as urban a nd hi ghwa y scenar ios. F or example, note that the proposed band i s close to the 5.9 GHz V2V band [21]. Howe ver , the differen ce betwee n 5.4 and 5.9 GHz is 9.3% ( = 0.5/5.4); thus, their propagation channel charac teristi cs do not change s igniﬁca ntly . Based on the measurements in [38] and [39], the pat h loss e xponent has a variat ion of less than 15% ov er 1 GHz bandwidt h and the delay spread has less than 10% v ariat ion ove r 8 GHz bandwidth. Here, we could re gard these v alues as an uncerta inty of the estimate d model parameters at 5.4 GHz, w hen the goal is to estimate paramete r valu es at 5.9 GHz. commun ication environmen ts by a d justing the model p aram- eters. Additionally , our model is time-variant because of the relativ e mo tion b etween the MT and MR. Consequ ently , we can analyz e the propo sed chann el statistics for more moving directions, rather than som e special moving conditions as mentioned in [28] . Further more, in the pro p osed model, the effect of road width on th e V2V chann el statistics can be in vestigated. I t is impo rtant to analyze the p roposed channel statistics fo r different taps and d ifferent p ath delay s in n on- stationary cond itions. T h is mod el further corrects the u nre- alistic assumption wid e ly used in current V2V RS-GBSMs. For exam p le, the authors in [5] ad opted the WSS c h annel to describ e the V2 V scena r ios; the imp act of non-station ary on V2V channel statistics was neglected. It is assumed that the azimuth angle of d eparture (AAoD), elev atio n an gle of departur e (EAoD), azim uth angle of arriv al (AAoA) , and elev ation ang le of a rriv al (E AoA) a re indepen dent of e ach other [25]. The major contributions of this paper are outlined as follows: (1) Based on the two measured scenarios mentioned above in [20 ], we pro pose a 3D non-station ary wideband geometric channel mo del for two different V2V co mmunicatio n environ- ments, i.e., high way scenarios and urban scenarios. (2) W e outlin e th e statistical pr operties of the pro posed V2V channe l model for different taps. Im portant time-variant channel statistics are derived and tho rough ly invest igated. Speciﬁcally , the time-variant space an d frequen cy c orrelation function s (CFs) and co rrespond ing Dop p ler power spectral densities ( PSDs) are de riv ed for V2V scen arios with different relativ e moving directions. (3) Th e impacts of non -stationarity (i.e., relativ e moving time and relativ e moving d irections) on time-variant space and frequen cy CFs are in vestigated in a compar ison with those of th e cor respondin g WSS mod el and measured r esults. The results sh ow that th e prop osed chann el mod e l is an excellent approx imation o f the realistic V2V scenarios. (4) The geo metric pa th length s betwee n the MT an d MR in a 3D semi-ellipsoid V2V chan nel m o del continu e to chan ge because th e tra nsmit az imuth an d ele vation angles constantly vary . W e thus analyze th e pr o posed statistical prop erties for different taps an d different path delays, which is a different approa c h than those pr esented in previous works [12,25,2 7]. The remain d er of this p aper is organized as f ollows. Section II details the p roposed theoretical 3D no n-stationary wideband MIMO V2V ch annel model. I n Section II I, based o n the pro- posed geo metric model, the time-variant spa c e CFs, freq uency CFs, and corr esponding Doppler PSDs ar e derived. Numerical results and d iscussions are pr ovided in Section IV . Finally , ou r conclusion s are presented in Section V . I I . 3 D G E O M E T RY - B A S E D V 2 V T H E O R E T I C A L C H A N N E L M O D E L In V2V scenarios, the impacts of moving vehicles and roadside en viro n ments o n th e ch annel statistical properties should be add r essed [11,21 ]. Additionally , the relativ e move- ment between the MT and MR m akes the V2V cha n nel time- variant. Howe ver, the p revious chan nel models have certain 3 ] [ \ ]  \   W  W  < = x  x   a   a  b  b Fig. 1. Proposed 3D wideband MIMO V2V channel model combining the two-c ylinder model and multiple confocal semi-ell ipsoid models for the line- of-sight (LoS) propagation rays. ] [ ]  \  \ 07 05 Fig. 2. Geometric angle s and path lengths of the proposed V2V channel model for s ingle- and double-bounce d pro pagati on rays. limitations in term s of realistically describin g th e V2V com- munication environments. For example, the mod els in [1 0] and [14 ] rely on th e WSS assump tio n, wh ich implies th at in the time dom ain, the statistical prop erties o f the chan nel remain inv ar iant over a short pe r iod of time. Th u s, the above channel m odels co uld not d e pict the rea l V2V environments because of the motion b etween the MT an d MR. T h e authors in [15] and [19 ] presen ted the semi-ellip soid and cylind er models, respectively , to describe the scatterers surrou n ding the transmitter and r eceiv er . Howe ver, in these studies, th e effect o f the roadside en v ironmen ts on the chann el charac ter istics was not discussed. In [3 1] and [32], the a uthors p roposed e llipsoid channel mod els to d escribe the m o bile radio en vironm e n ts. Howe ver, the moving veh ic le s arou nd the MT and MR were not investigated in V2V environments. On the other han d, the authors in [28] perform ed the channel measurements on ly with the MT and M R dr iving in the same d irection, and with th e MT and MR driving in op posite direction s. Howe ver , the effect of th e arb itrary movin g direction s on the ch annel statistics was n ot investigated. Motiv ated by the above d rawbacks, we have adopted a 3D non- statio n ary wideb a nd geom etric channel m o del in th is paper to depic t the ac tual vehicular commun ications, as illustrated in Figs. 1 an d 2. In the prop osed channel model, we assume that th e MT a n d MR are lo c ated in the same azimuth plane. Th us, the mod el is mainly applica b le f or ﬂat road condition s. Similar assumption can be seen in [25] and [27]. Howe ver, in reality , the vehicles can b e anyw h ere above, b elow , o r on the actual slope, requ iring a m ore car e ful analysis to accu rately mo del this V2V prop a - gation co ndition. For example, the author s of [ 3 5] presented path loss channe l mo dels for slop ed-terrain scenario s, in wh ich the ground reﬂec tion w a s considered in the V2V channels. I n this study , the author s introdu ced four V2V scen arios: (1) two vehicles are located at op posite end s o f the slope; (2) one vehicle is o n the slope, and the oth er vehicle is beyond the slope cr est; ( 3) o ne vehicle is on the slope, and the other is away from th e slope at the bottom; (4 ) bo th vehicles ar e on the slope. Figs. 1 and 2 illustrate the g eometry of the prop osed V2V cha n nel model, wh ich is the co mbination of line-of -sight (LoS), single-, and doub le-boun ced propag ation rays. H e re, we u se a two-cylinder mode l to depict moving vehicles (i.e., around the MT o r MR). W e employ multiple conf ocal semi- ellipsoid mo dels to mimic station ary ro adside environments. In gener al, we note that most struc tures in macr ocell scenarios (e.g., buildings, high ways, urban spaces) have straight vertical surfaces. Thus, we adop t vertical cylinders to m odel the scattering surfaces rep resented by movin g vehicles [13 , 19]. Because the heig hts of the vehic les and ped estrians are similar to th ose o f the tran smitter and receiver , we can assume th at the scatterers lie on the cylinder model at th e MT an d M R in the propo sed 3 D space. T o justify this assum p tion, co rrespond ing compariso n s are made between the assump tions o f the moving vehicles of the 2D circle and 3D c ylinder mod els. The results show that the power levels of th e Do ppler sp ectrum between these models are insign iﬁcant. Additionally , we introdu c e the 3D semi-ellipsoid mode l because of the fo llowing poin ts. (1) For V2V commun ications, it is acce p table to intr oduce an ellipse chan nel mo del with an MT and MR lo cated at the foci to de scribe th e roadside environments [5]; howe ver , they n eglect the transmission signal in the vertical plan e. (2) Geometric p ath length s b etween the MT and MR in a 3D semi-ellipsoid V2V channel mo del con tinue to ch ange as the transmit a z imuth and elevation angles co nstantly vary . Thus, we c a n analy ze the pro posed statistical pro perties for different path delays as the tap is ﬁxed. (3) W e can f urther analyze the channel statistics for different path delays in different taps. This appr oach is signiﬁcantly different from those in p revious works [ 1 2,27] . T o the b est of ou r knowledge, this is the ﬁrst time th at a 3 D semi- e llipsoid model is used to m imic V 2 V channels. As shown in Figs. 1 and 2, supp ose tha t the MT and MR are eq uipped with un iform linear array (UL A ) M T and M R omnidire ctional a n tenna elem e nts. The propo sed mo del is also capab le o f introduc ing o th er MIM O geome tr ic an- tenna sy stem s, such as uniform circular array (U CA), unif o rm rectangu la r array (URA), an d L-shap ed array . The d istan ce between th e cen ters o f the MT and MR cylinder s are denoted as D = 2 f 0 , where f 0 designates the half- length of th e d istance between the two focal points of the ellipse. Let us deﬁne a l , b l , and u l as the semi-major axis of the thre e d im ensions of the l th semi-ellipsoid, where b l = p a 2 l − f 2 0 . It is assumed tha t 4 07 05 B /R6  W C A 07 05 M G E F N U V   a   b  W  W l W  W l W Fig. 3. The ellipse model describing the path geometry (a) ﬁrst tap; (b) other taps. the radius of the c ylindrical surface a r ound th e MT is den oted as R t 1 ≤ R t ≤ R t 2 . Note that R t 1 and R t 2 correspo n d with the respective ur b an and h ighway scenarios in [20]. Similarly , at th e MR, the radius of the cylind rical surface is den oted as R r 1 ≤ R r ≤ R r 2 . Let Ant T p represent the p th ( p = 1 , 2 , ..., M T ) antenna o f the transmit array , and let Ant R q represent th e q th ( q = 1 , 2 , ..., M R ) antenn a of the receive array . The spaces between the two adjacen t antenna elements at th e MT and MR are deno ted as δ T and δ R , respectively . The or ientations of the tran smit an tenna arr ay in the azimuth pla n e (relative to the x -axis) an d elev ation plan e ( r elativ e to the x-y plane ) ar e denoted as ψ T and θ T , respectively . Similarly , the orientations at th e receiver are d enoted as ψ R and θ R , r espectiv ely . Here , we assume that ther e are N 1 , 1 scatterers (moving vehicles) existing on the cylindrica l surface arou nd the MT , an d the n 1 , 1 th ( n 1 , 1 = 1 , ..., N 1 , 1 ) scatterer is deﬁned as s ( n 1 , 1 ) T . N 1 , 2 effecti ve scatterers likewise exist ar ound the MR lying on the cylind e r model, and the n 1 , 2 th ( n 1 , 2 = 1 , ..., N 1 , 2 ) scatterer is deﬁned as s ( n 1 , 2 ) R . For th e m ultiple con focal semi-ellipsoid m o dels, N l , 3 scatterers lie o n a multiple c o nfocal semi-ellipsoid with th e MT and M R located at the foc i. T he n l , 3 th ( n l , 3 = 1 , ..., N l , 3 ) scatterer is d esignated as s ( n l , 3 ) . Although the propo sed cha nnel model only takes into acco unt th e a zimuth and elev a tio n ang les in th e 3 D spac e , it can also be used in polar ized anten na arrays [40], as the polarization angles are co nsidered in the m o del. In multipath cha n nels, the path length of each wave deter- mines the propag ation delay a n d essentially also the average power of the wave at th e MR. In [21], the au thors state th at the ellipse mode l forms to a certain extent the ph y sical basis for th e m o delling of freq uency-selective channe ls. Theref o re, when the MT and MR are located in th e focus of the ellipse, ev ery wave in the scatterin g region characterized b y th e l th ellipses u n dergoes the same discrete pro pagation delay τ ℓ = τ 0 + ℓτ , ℓ = 0 , 1 , 2 , ..., L − 1 , where τ 0 denotes the propag ation delay o f th e L oS comp onent, τ is an in ﬁnitesimal propag ation d e lay , and L is th e n umber of paths with different propag ation delays. In pa r ticular , the numb er of paths ℓ with different pro pagation delay s exactly cor respond s to the numbe r of delay elem e n ts r equired for the tapp ed-delay -line (TDL) structure of mo delling f requen cy-selective chann els. W e ob- serve that in r eal V2V com munication scenarios with different contributions of single- and d ouble-b ounced rays to the V2V channel statistics, it is necessary to design different taps of the pro posed wideba n d V2V cha n nel model. As mentioned in [36], the tap is strong ly related to the delay resolutio n in V 2 V channels. He r e, let us deﬁne a l as th e semi-m ajor of the l th ellipse in the azimu th plane . Then, for th e n ext time de la y , the semi-major o f the ( l + 1) th ellipse in the azimuth plan e can be derived as a l +1 = a l + cτ / 2 with c = 3 × 1 0 8 m/s. Modelling V2V channels by using a TDL stru c tu re with time-variant co efﬁcients gives a deep insight into the channel statistics in the propo sed model. In Fig . 3(a) , we notice that the received signal for the ﬁrst tap is composed of an inﬁnite number o f d elayed and weigh ted replica s of the transmitted signal in a multipath c hannel, inclu ding direct LoS r ays (i.e., MT → MR), single- bounced r a ys caused b y the scatterers located on e ither of the two cylinders (i.e. , MT → A → M R and MT → B → MR) or o n the ﬁrst semi-ellip soid (i.e., MT → C → MR), and d ouble-b ounced ray s genera ted from the scatterers located o n both cylinders (i.e., MT → U → V → MR). Here, let u s deﬁne the comb ination o f the above cases as the ﬁrst tap. Thus, we c a n analyz e the propo sed channel characteristics for different time delays, i.e., per-tap channel statistics, which is meaningfu l for V2V ch annels. Howev er , for o ther tap s ( l ≥ 1 ), the link is a sup erposition of th e sing le- bounc e d ray s th at ar e p roduc e d on ly fr om the scatterers located on the corr espondin g semi-ellipsoid (i.e. , MT → G → MR), as well as the dou b le-boun ced rays caused by the scatterers fro m the comb ined single cylind er (i.e., MT → E → F → MR) and th e corre sp onding semi-ellipsoid (i.e., MT → M → N → MR), as shown in Fig. 3 (b). In g eneral, the pr oposed V2V channel model can be de- scribed by matr ix H ( t ) =  h pq ( t , τ )  M T × M R of size M T × M R . Therefo re, the complex imp ulse re sp onse betwee n the p th transmit anten na and q th rec ei ve antenna in our model can be expressed as h pq ( t , τ ) = P L ( t ) l =1 ω l h l ,pq ( t ) δ ( τ − τ l ( t )) , wh ere the subscr ip t l r epresents th e tap numb er , h l ,pq ( t ) denotes th e complex tap coe fﬁcient of the Ant T p → Ant R q link, L ( t ) is the total num ber of taps, ω l is the attenu ation factors of the l th tap, and τ l is the cor respond in g prop agation time delays [2 5]. A. Pr opo sed 3D channel model description Based on the ab ove analysis, the comp lex tap coe fﬁcient for the ﬁrst tap o f the Ant T p → Ant R q link at the carrier frequen cy f c can be expr essed as [13][ 27] h 1 ,pq ( t ) = h LoS 1 ,pq ( t ) + 3 X i =1 h SB 1 , i 1 ,pq ( t ) + h DB 1 ,pq ( t ) (1) 5 with h LoS 1 ,pq ( t ) = r Ω Ω + 1 e − j 2 πf c ξ pq / c + j 2 π t × f max cos  α LoS R − γ R  cos β LoS R (2) h SB 1 , i 1 ,pq ( t ) = r η SB 1 , i Ω + 1 lim N 1 , i →∞ N 1 , i X n 1 , i =1 1 √ N 1 , i e − j 2 π f c ξ pq,n 1 , i / c × e j 2 π t × f max cos  α ( n 1 , i ) R − γ R  cos β ( n 1 , i ) R , i = 1 , 2 , 3 . (3) h DB 1 ,pq ( t ) = r η DB Ω + 1 × lim N 1 , 1 , N 1 , 2 →∞ N 1 , 1 , N 1 , 2 X n 1 , 1 , n 1 , 2 =1 s 1 N 1 , 1 N 1 , 2 × e − j 2 π f c ξ pq,n 1 , 1 ,n 1 , 2 / c × e j 2 π t × f max cos  α ( n 1 , 2 ) R − γ R  cos β ( n 1 , 2 ) R (4) where ξ pq,n 1 , i = ξ pn 1 , i + ξ qn 1 , i and ξ pq,n 1 , 1 ,n 1 , 2 = ξ pn 1 , 1 + ξ n 1 , 1 n 1 , 2 + ξ qn 1 , 2 denote the trav el distance o f the waves through the link Ant T p → s ( n 1 , i ) → Ant R q and Ant T p → s ( n 1 , 1 ) T → s ( n 1 , 2 ) R → An t R q , respectively . Her e, Ω de notes the Rice factor an d f max is the maximum Doppler fr equency with respect to the MR [11]. α LoS R and β LoS R denote the AAoA and EAo A of the LoS path, respectively . For th e NLoS r ays, the symbol α ( n 1 , 1 ) R represents the AAoA of the wav e scatter ed from the effecti ve scatterer s ( n 1 , 1 ) T around the MT , whereas α ( n 1 , 2 ) R represents the AAoA of the wav e scattered fro m the scatterer s ( n 1 , 2 ) R around the MR. Similarly , β ( n 1 , 1 ) R and β ( n 1 , 2 ) R denote the EAoAs o f the waves scattered from the scatter er s ( n 1 , 1 ) T and s ( n 1 , 2 ) R , respecti vely . On the other hand, α ( n 1 , 3 ) R and β ( n 1 , 3 ) R denote the AAoA and EAoA of the waves scattered from th e scatterer s ( n 1 , 3 ) in the semi- ellipsoid model for the ﬁrst tap. It is evident th at the MT and MR are bo th in motion, we h erein assum e that the MR m oves in a r elati ve direction to the MT with th e p rinciples o f relativ e motion. 2 Similar work can be seen in [2] and [5 ]. In this case, different chan nel char acteristics can be de scr ibed by adjusting the re lated mo del param eters. Here, we assume that the MR moves in an arbitrar y direction , γ R , with a constant velocity of v R at time instant t in the azimuth p lan e. Furtherm ore, energy-related parameters η SB 1 , i and η DB specify the numbers of the sing le- and dou ble-bou nced r a y s respectively contribute to th e total scatter ed power , wh ich can be no r malized to satisfy P 3 i =1 η SB 1 , i + η DB = 1 fo r b revity [11,2 1]. Howe ver , as shown in Fig. 2, for other taps ( l ≥ 1), the com plex tap coefﬁcient of the An t T p → Ant R q link can b e derived as h l ,pq ( t ) = h SB l , 3 l ,pq ( t ) + h DB l , 1 l ,pq ( t ) + h DB l , 2 l ,pq ( t ) (5) with h SB l , 3 l ,pq ( t ) = √ η SB l , 3 lim N l , 3 →∞ N l , 3 X n l , 3 =1 1 √ N l , 3 e − j 2 π f c ξ pq,n l , 3 / c × e j 2 π t × f max cos  α ( n l , 3 ) R − γ R  cos β ( n l , 3 ) R (6) 2 Although the e xisting channel models, where the MT and MR are both in motion with same or opposite directions, seem more reasonable to reﬂect the actua l vehi cular en vironment s. In reality , these models cannot study the effec ts of arbitrar y moving directi ons on the channel statistics, which are meaningful for V2V channel. h DB l , 1 l ,pq ( t ) = √ η DB l , 1 lim N 1 , 1 , N l , 3 →∞ N 1 , 1 , N l , 3 X n 1 , 1 , n l , 3 =1 s 1 N 1 , 1 N l , 3 × e − j 2 π f c ξ pq,n 1 , 1 ,n l , 3 / c × e j 2 π t × f max cos  α ( n l , 3 ) R − γ R  cos β ( n l , 3 ) R (7) h DB l , 2 l ,pq ( t ) = √ η DB l , 2 × lim N l , 3 , N 1 , 2 →∞ N l , 3 , N 1 , 2 X n l , 3 , n 1 , 2 =1 s 1 N l , 3 N 1 , 2 × e − j 2 π f c ξ pq,n l , 3 ,n 1 , 2 / c × e j 2 π t × f max cos  α ( n 1 , 2 ) R − γ R  cos β ( n 1 , 2 ) R (8) where ξ pq,n l , 3 = ξ pn l , 3 + ξ qn l , 3 , ξ pq,n 1 , 1 ,n l , 3 = ξ pn 1 , 1 + ξ n 1 , 1 n l , 3 + ξ qn l , 3 , a n d ξ pq,n l , 3 ,n 1 , 2 = ξ pn l , 3 + ξ n l , 3 n 1 , 2 + ξ qn 1 , 2 denote the travel distance o f th e waves throug h the link An t T p → s ( n l , 3 ) → An t R q , Ant T p → s ( n 1 , 1 ) T → s ( n l , 3 ) → Ant R q , and Ant T p → s ( n l , 3 ) → s ( n 1 , 2 ) R → Ant R q , respecti vely . α ( n l , 3 ) R and β ( n l , 3 ) R denote the AAoA and EAoA of the waves scattered from the scatterer s ( n l , 3 ) in th e l th semi-ellipsoid mo del for other tap s. Similar to the ab ove case, en ergy-related param eters η SB l , 3 and η DB l , 1 ( η DB l , 2 ) specify the amount th at th e single- an d do uble-bo unced rays respectively co ntribute to the to tal scattered power , wh ich can be normalized to satisfy η SB l , 3 + η DB l , 1 + η DB l , 2 = 1 for brevity . In addition, because th e deriv atio ns of the cond ition that gu arantees the f ulﬁllment of th e TDL structure are the same, we on ly detail the deriv ation of the condition f or the second tap. As intro duced in [12] and [2 7], we note tha t the impulse response of the propo sed mod e l is relate d to the scatter e d power in V2V chann e ls. T h erefore , it is importan t to d eﬁne the r e ceiv ed scattered power in d ifferent taps and different V2V scenarios (i.e., highway an d urban scenarios) in the propo sed non-stationary channel mo del. In sh ort, fo r the ﬁrst tap, the single-bo unced rays are caused by the scatterers located o n either of the two cylinders or the ﬁrst semi- ellipsoid, while th e dou b le-boun ced ra y s are gener ated from the scatterers located on th e b oth cylinder s, as shown in Fig. 2. For highway scenar ios (i.e., R t = R t 2 and R r = R r 2 ), the higher relative movement of the vehicles results in a h igher Doppler frequen cy; moreover , the value of Ω is always large because the LoS compon ent can bear a signiﬁcant amou nt of power . Additionally , the r eceiv ed scattered power is mainly from wa ves reﬂected by the stationary roadside en vir o nments described by the scatterer s located on the ﬁr st semi-ellipsoid. The moving vehicle s represented by the scatterers located on the two cylinders are mo re likely to be single- b ounced , rather th a n doub le-boun ced. T his ind icates that η SB 1 , 3 > max { η SB 1 , 1 , η SB 1 , 2 } > η DB . For urban scenarios (i.e., R t = R t 1 and R r = R r 1 ), the lower relative movemen t of th e vehicles results in a lower Doppler frequency; mor eover , the v alue of Ω is smaller th a n that in the h ighway scenarios. Addition a lly , the double- bounc e d ray s of th e two-cylinde r mod el can bear mor e energy than the single-bou nced r ays of th e two-cylinder a nd semi-ellipsoid models, i.e., η DB > max { η SB 1 , 1 , η SB 1 , 2 , η SB 1 , 3 } . 6 \ [ 07 05 05 Fig. 4. T op view of the geometric angles and path length s in the proposed non-stati onary V2V channel model. Howe ver, for the secon d tap, it is assumed tha t the single- bounc e d r ays are produced only from the static scatter ers located on th e corre sponding sem i-ellipsoid, while the doub le- bounc e d rays are caused by the scatterer s from the combined one cylinder (eith er o f th e two cylinders) and the correspond- ing semi-e llip soid [21 ,26,27 ]. Note that, in th e p r oposed TDL structure, th e double- bounc e d rays in the ﬁrst tap must be smaller in distan ce than the single-b ounced r ays on the second semi-ellipsoid, i.e., max { R t , R r } < min { a 2 − a 1 } . It is stated in [ 41] that the dela y resolution is app roximately the inverse of bandwidth and therefore, we a ssum e that th e delay resolution in the pro posed m odel is 2 0 ns for 50 MHz. In this pap er , we deﬁne d ifferent time d e lays with th e different ellipses. Thus, the second ellipse should produ ce at least 6 m excess path length than the ﬁrst ellipse, i.e., 2 a 2 − 2 a 1 = c τ with τ = 20 ns. In this case, the prop o sed chan nel statistics for different time delays, i.e., per-tap statistics, can be inv estigated. For hig hway scenario s, the received scattered p ower is mainly from wa ves reﬂected by the stationary roadside en vironme n ts described by the scatter ers located on th e semi-ellipsoid, i.e. , η SB l , 3 > max { η DB l , 1 , η DB l , 2 } . For urban scen arios, the do uble- bounc e d rays from the combined sin g le cylinder and semi- ellipsoid mode ls can bear more en ergy than the single-boun ced rays o f the semi-ellip so id mo del, i.e., min { η DB l , 1 , η DB l , 2 } > η SB l , 3 . B. Non-station ary time-variant parameters T o describe the non- stationarity o f the pr o posed 3D wide- band cha n nel model, we introd uce a V2V c o mmunica tion sce- nario, as illustrated in Fig. 4. T he ﬁgure sh ows the geom etric proper ties and moving statistics of the proposed model in the azimuth p la n e. In this ca se, owing to overly com plex issues, the cor respondin g 3 D ﬁgure with MIM O antennas is omitted for brevity . For V2V scenario s, the geom e tric paths len g ths will be time-variant becau se of the relative movement between the MT and M R. Co n sequently , ξ pq , ξ pn 1 , 2 , ξ qn 1 , 1 , ξ qn l , 3 , an d ξ n 1 , 1 n 1 , 2 can b e rep laced by ξ pq ( t ) , ξ pn 1 , 2 ( t ) , ξ qn 1 , 1 ( t ) , ξ qn l , 3 ( t ) , and ξ n 1 , 1 n 1 , 2 ( t ) , resp e c ti vely . Howe ver , in Fig. 4, no te that th e distances ξ pn 1 , 1 , ξ pn l , 3 , and ξ qn 1 , 2 have no related to the non- stationary properties, i.e., ξ pn 1 , 1 ( t ) = ξ pn 1 , 1 , ξ pn l , 3 ( t ) = ξ pn l , 3 , and ξ qn 1 , 2 ( t ) = ξ qn 1 , 2 . In g eneral, it is clear ly observed that the MR is relatively far f r om th e MT in th e pr oposed V2V co mmunicatio n e nvironmen ts. Thus, we can make the following assumptions: min { R t , R r , u − f } ≫ ma x { δ T , δ R } , D ≫ max { δ T , δ R } , and the appro ximation √ x + 1 ≈ 1 + x / 2 is used for small x . Accord ingly , b a sed o n the law of cosines in appropr iate triang les and small ang le appr oximation s (i.e., sin x ≈ x an d cos x ≈ 1 fo r small x ) [1 2 ,27], the co rrespon ding time-variant geometr ic path lengths at r elati ve moving time instant t can be a pprox im ated as ξ pq ( t ) ≈ q  D − δ Tx  2 +  v R t  2 − 2  D − δ Tx  v R t cos  α LoS R − γ R  (9) ξ pn 1 , 1 ( t ) ≈ R t − h δ Tx cos α ( n 1 , 1 ) T cos β ( n 1 , 1 ) T + δ T y sin α ( n 1 , 1 ) T cos β ( n 1 , 1 ) T + δ Tz sin β ( n 1 , 1 ) T i (10) ξ pn 1 , 2 ( t ) ≈   D − Q p cos θ T  2 +  v R t  2 − 2  D − Q p cos θ T  v R t cos  α ( n 1 , 2 ) R − γ R   (11) ξ pn l , 3 ( t ) ≈ 2 a 2 l b 2 l u 2 l ξ l, 3 − δ T cos θ T h R r / D sin ψ T sin α ( n l , 3 ) T + cos ψ T i (12) ξ qn 1 , 2 ( t ) ≈ R r − h δ Rx cos α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R + δ Ry sin α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R + δ Rz sin β ( n 1 , 2 ) R i (13) ξ qn 1 , 1 ( t ) ≈   D − Q q cos θ R  2 +  v R t  2 − 2  D − Q q cos θ R  v R t cos  α ( n 1 , 1 ) R − γ R   (14) ξ qn l , 3 ( t ) ≈  ξ 2 pn l , 3 ( t ) sin 2 β ( n l , 3 ) T + ξ 2 R +  v R t  2 + 2 D ξ R cos  γ R + α ( n l , 3 ) R   (15) ξ n 1 , 1 n 1 , 2 ( t ) ≈ q D 2 +  v R t  2 − 2 Dv R t cos  α LoS R − γ R  (16) where δ Tx = δ T cos θ T cos ψ T , δ T y = δ T cos θ T sin ψ T , δ Tz = δ T sin θ T , δ Rx = δ R cos θ R cos ψ R , δ Ry = δ R cos θ R sin ψ R , δ Rz = δ R sin θ R , Q p = δ T cos θ T  R r / D sin ψ T sin α ( n 1 , 2 ) R + cos ψ T  , ξ l, 3 = b 2 l u 2 l cos 2 β ( n l , 3 ) T cos 2 α ( n l , 3 ) T + a 2 l u 2 l cos 2 β ( n l , 3 ) T sin 2 α ( n l , 3 ) T + a 2 l b 2 l sin 2 β ( n l , 3 ) T , ξ R =  D 2 + ξ 2 pn l , 3 ( t ) cos 2 α ( n l , 3 ) T − 2 D ξ pn l , 3 ( t ) cos β ( n l , 3 ) T cos α ( n l , 3 ) T  , and Q q = δ R cos θ R  R t / D sin ψ R sin α ( n 1 , 1 ) T − cos ψ R  . T o jointly con sid er th e impact of the azimu th and elevation angles on chann el statisti cs, several scatterer d istributions, 7 such as uniform , Gaussian, Laplacian, and von Mises, were used in pr ior w ork. Here, we adopt the v o n M ises pro bability density func tio n (PDF) to c haracterize the distribution of scatterers in the pr oposed V2V channel. Thus, the von Mises PDF is d eriv ed as p ( α ( n l , i ) R , β ( n l , i ) R ) = k cos ( n l , i ) R 4 π sinh k × e k cos β 0 cos β ( n l , i ) R cos  α ( n l , i ) R − α 0  × e k sin β 0 sin β ( n l , i ) R (17) with α ( n l , i ) R and β ( n l , i ) R ∈ [ − π , π ) , α 0 ∈ [ − π , π ) . In add ition, β 0 ∈ [ − π , π ) den otes the m e a n values of the azimuth angle α ( n l , i ) R and elev atio n angle β ( n l , i ) R at the receiver , respec ti vely . In addition, k ( k ≥ 0) is a r eal-valued param eter that con trols the ang les spread of α 0 and β 0 [12]. As previously mentio ned, different ch annel ch aracteristics can be d escribed b y adju sting the pr oposed model paramete r s. For exam p le, it is appar e nt that wh en we do no t take the roadside environments into accou nt, the p r oposed mode l ten ds to the Zajic model [13,1 9 ]. Howe ver, the prop osed mod el is suitable for the previous 3D stationary semi-ellipso id channels as t = 0 , as shown in [31] and [32]. In this case, our channel can be degenerated into a 2D elliptical channel as model parameter u l is equ al to zero. On the other han d, whe n we set t 6 = 0 , our model can can a lso be used to depict non-station ary V2V channels, suc h as the Ghazal model [26] and Y u an model [27]. Like wise, the pro posed mod el describes other mod els in previous w ork; we o mit them for br evity . I I I . P R O P O S E D T H E O R E T I C A L M O D E L S T A T I S T I C A L P R O P E RT I E S A. Space CFs In genera l, CF is an importan t statistic in d esigning com- munication link that ch aracterizes how fast a wireless channe l changes with respect to time, movement, or frequen cy [25]. The speciﬁc CF that is o f inter e st in this paper is the space CF , which m e a sures th e spatial statistics of the proposed V2V channel. It is stated in [27] th at the spatial correlation p r op- erties o f two arbitra r y channel imp ulse responses h pq ( t ) and h p’q’ ( t ) of a MI MO V 2 V channel ar e co mpletely dete r mined by the cor relation pro perties of h l, pq ( t ) and h l, p’q’ ( t ) in each tap, so th a t no co rrelations exist between the u nderlyin g pr ocesses in different tap s. Theref ore, th e norm alized time- variant space CF can be expr essed as [36] ρ h l ,pq , h l ,p’q’  t , τ  = E h h l ,pq  t  h ∗ l ,p’q’  t + τ  i r E h   h l ,pq ( t )   2 i E h   h l ,p’q’ ( t + τ )   2 i (18) where ( · ) ∗ denotes the complex co njugate oper ation and E [ · ] is the expectation operation. Because the Lo S, single-, and d ouble-b ounced ray s are independ ent o f each o ther, the channel respon se fo r the ﬁrst tap c a n be expressed as ρ h 1 ,pq , h 1 ,p’q’  t , τ  = ρ LoS h 1 ,pq , h 1 ,p’q’  t , τ  + 3 X i =1 ρ SB 1 , i h 1 ,pq , h 1 ,p’q’  t , τ  + ρ DB h 1 ,pq , h 1 ,p’q’  t , τ  (19) Howe ver, for other taps, accord ing to ( 1 8), we have the time-variant space CF as ρ h l ,pq , h l ,p’q’  t , τ  = ρ SB l , 3 h l ,pq , h l ,p’q’  t , τ  + ρ DB l , 1 h l ,pq , h l ,p’q’  t , τ  + ρ DB l , 2 h l ,pq , h l ,pq  t , τ  (20) By applyin g th e correspond ing scatterer non- unifor m dis- tribution, and by following similar reasoning in [19,2 1], we can o btain the time - variant space CFs o f the Lo S, single - , and double-b ounced rays, as outlined below . Sp e ciﬁcally , by submitting (2) into (18), the time-variant space CF in the case of the Lo S rays can be expr essed as ρ LoS h 1 ,pq , h 1 ,p’q’  t , τ  = K e j 2 πf c c r  D − δ Tx  2 +  v R t  2 − 2  D − δ Tx  v R t cos  α LoS R − γ R  × e j 2 πf max τ cos  α LoS R − γ R  (21) where λ denotes the wavelength. In submitting (3 ) in to ( 18), the time-variant space CF in the case of th e sing le-boun ced rays SB 1 , i can be derived as ρ SB 1 , i h 1 ,pq , h 1 ,p’q’  t , τ  = η SB 1 , i lim N 1 , i →∞ N 1 , i X n 1 , i =1 1 N 1 , i e j 2 πf c c A ( SB 1 , i ) + B ( SB 1 , i ) (22) where A ( SB 1 , 1 ) =  ( D − Q q cos θ R ) 2 + ( v R t ) 2 − 2( D − Q q cos θ R ) × v R t co s( α ( n 1 , 1 ) R − γ R )  − δ Tx cos α ( n 1 , 1 ) T cos β ( n 1 , 1 ) T − δ T y × sin α ( n 1 , 1 ) T cos β ( n 1 , 1 ) T − δ Tz sin β ( n 1 , 1 ) T , B ( SB 1 , 1 ) = j 2 πf max τ co s( α ( n 1 , 1 ) R − γ R ) cos β ( n 1 , 1 ) R , A ( SB 1 , 2 ) =  ( D − Q p cos θ T ) 2 + ( v R t ) 2 − 2  D − Q p  × v R t co s( α ( n 1 , 2 ) R − γ R )  1 / 2 − δ Rx cos α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R − δ Ry sin α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R − δ Rz sin β ( n 1 , 2 ) R , and B ( SB 1 , 2 ) = j 2 π f max τ co s( α ( n 1 , 2 ) R − γ R ) × cos β ( n 1 , 2 ) R . It is stated in [6] that P N 1 , i n 1 , i =1 1 / N 1 , i = 1 as N 1 , i → ∞ . Thus, the to tal power o f the SB 1 , i rays is proportion a l to 1 / N 1 , i . This is equal to the inﬁnitesimal power com ing fr om the differential of the 3D ang les, d α ( n 1 , i ) R d β ( n 1 , i ) R , i.e., 1 / N 1 , i = p ( α ( n 1 , i ) R , β ( n 1 , i ) R ) d α ( n 1 , i ) R d β ( n 1 , i ) R , wher e p ( α ( n 1 , i ) R , β ( n 1 , i ) R ) d e- notes the joint von Mises PDF in (17). Th erefore , (22) can be rewritten as ρ SB 1 , i h 1 ,pq , h 1 ,p’q’  t , τ  = η SB 1 , i Z π − π Z π − π e j 2 πf c c A ( SB 1 , i ) + B ( SB 1 , i ) × p  α ( n 1 , i ) R , β ( n 1 , i ) R  d α ( n 1 , i ) R d β ( n 1 , i ) R (23) Similarly , submitting (4) into (18) , the time-variant space CF in the case of the single-bounce d rays SB l , 3 can be expressed as 8 ρ SB l , 3 h l ,pq , h l ,p’q’  t , τ  = η SB l , 3 Z π − π Z π − π e j 2 πf c c A ( SB l , 3 ) + B ( SB l , 3 ) × p  α ( n l , 3 ) R , β ( n l , 3 ) R  d α ( n l , 3 ) R d β ( n l , 3 ) R (24) where A ( SB l , 3 ) =  ξ 2 pn l , 3 ( t ) sin 2 β ( n l , 3 ) T + ξ 2 R + ( v R t ) 2 + 2 D × ξ R cos( γ R + α ( n l , 3 ) R )  − δ Tx cos θ T  R r / D sin ψ T sin α ( n l , 3 ) T + cos ψ T  and B ( SB l , 3 ) = j 2 π τ cos( α ( n l , 3 ) R − γ R ) cos β ( n l , 3 ) R . Submitting ( 6) in to (1 8), the time- variant spac e CF in the case of the double- bounc e d rays DB can be expressed as ρ DB h 1 ,pq , h 1 ,p’q’  t , τ  = η DB Z π − π Z π − π e j 2 πf c c A ( DB ) + B ( DB ) × p  α ( n 1 , 2 ) R , β ( n 1 , 2 ) R  d α ( n 1 , 2 ) R d β ( n 1 , 2 ) R (25) where A ( DB 1 , 2 ) = δ Tx cos α ( n 1 , 1 ) T cos β ( n 1 , 1 ) T + δ T y sin α ( n 1 , 1 ) T × cos β ( n 1 , 1 ) T + δ Tz sin β ( n 1 , 1 ) T + δ Rx cos α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R + δ Ry sin α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R + δ Rz sin β ( n 1 , 2 ) R and B ( DB 1 , 2 ) = j 2 πτ × cos( α ( n 1 , 2 ) R − γ R ) cos β ( n 1 , 2 ) R . Submitting ( 7) in to (1 8), the time- variant spac e CF in the case of the double- bounc e d rays DB l , 1 can be d eriv ed as ρ DB l , 1 h l ,pq , h l ,p’q’  t , τ  = η DB l , 1 Z π − π Z π − π e j 2 πf c c A ( DB l , 1 ) + B ( DB l , 1 ) × p  α ( n l , 3 ) R , β ( n l , 3 ) R  d α ( n l , 3 ) R d β ( n l , 3 ) R (26) where A ( DB l , 1 ) =  ξ 2 pn l , 3 ( t ) sin 2 β ( n l , 3 ) T + ξ 2 R + ( v R t ) 2 + 2 D × ξ R cos( γ R + α ( n l , 3 ) R )  − δ Tx cos α ( n 1 , 1 ) T cos β ( n 1 , 1 ) T − δ T y × sin α ( n 1 , 1 ) T cos β ( n 1 , 1 ) T − δ Tz sin β ( n 1 , 1 ) T and B ( DB l , 1 ) = j 2 π × f max τ cos( α ( n l , 3 ) R − γ R ) cos β ( n l , 3 ) R . Submitting ( 8) in to (1 8), the time- variant spac e CF in the case of the double- bounc e d rays DB l , 2 can be d eriv ed as ρ DB l , 2 h l ,pq , h l ,p’q’  t , τ  = η DB l , 2 Z π − π Z π − π e j 2 πf c c A ( DB l , 2 ) + B ( DB l , 2 ) × p  α ( n 1 , 2 ) R , β ( n 1 , 2 ) R  d α ( n 1 , 2 ) R d β ( n 1 , 2 ) R (27) where A ( DB l , 2 ) = δ Tx R r / D sin ψ T sin α ( n l , 3 ) T cos θ T + δ Tx cos θ T × cos ψ T + δ Rx cos α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R + δ Ry sin α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R + δ Rz sin β ( n 1 , 2 ) R and B ( DB l , 2 ) = j 2 π f max τ co s( α ( n 1 , 2 ) R − γ R ) × cos β ( n 1 , 2 ) R . From (21 )-(27 ), we no tice that th e time-variant space CFs are related no t only to the geo metric mo d el p arameters, but also to the moving prop erties. By substituting (17 ) in to (21)-( 27), the time- variant space cro ss-functions fo r the LoS, single-, and double-b ounced co mpone nts can be respectively obtained. Furth ermore, by setting p = p’ an d q = q ’ , the time- variant space auto-correlation f unction (A CF) can b e obtained [25]. On the other h and, we note that all above investigated statistical pro perties are time-variant on acco unt of th e non - WSS assump tion of the p r oposed V2V chann el m odel. Con- sequently , b y applying the Fourier transfo rmation o f h l ,pq ( t ) , the time-variant f requency cross-co rrelation f u nction of the propo sed 3 D non - stationary channel model can be d e r iv ed as ρ h l ,pq , h l ,p’q’ ( t , ∆ f ) = E h R ∞ −∞ h l ,pq ( t , τ ) h ∗ l ,pq ( t , τ ) e j 2 π ∆ f τ d τ i r E h   h l ,pq ( t , f )   2 i E h   h l ,p’q’ ( t , f + ∆ f )   2 i (28) Similar to the pr evious case, we substitute the co rrespond ing channel resp o nse into (2 8), and the time-variant f requency CFs for the LoS, sing le-, and do uble-bo unced p ropagatio n rays can be respectively d eriv ed. Nevertheless, the pr oposed channel model under the WSS assump tion (i.e., t = 0 ) demonstra te s that the channel statistics are not dependent on time t . In th is case, th e prop osed channe l model tends to be a conventional F2M channel m o del. Note that the above an alysis is mainly for the ﬂat com- munication environments, where the MR is far from the MT . Howe ver, when the MR is close to the MT , it is impo rtant to in vestigate the effect of groun d reﬂection on th e V2V ch annel statistics [35 ]. Here, we assume that th ere are N g effecti ve scatterers unifo r mly existing on the grou nd in the azimuth plane. The heigh ts o f a n tennas mounte d on th e MT and MR are den oted as H t and H r , respectively . The AAo A an d EAoA of the waves scattered from the scatterer on the grou n d are denoted as α ( n g ) R and β ( n g ) R , r espectiv ely . The distances from the MT and MR to the scatter er o n the groun d are deno ted as ξ pn g and ξ qn g , respectively . The ene rgy-related par ameter for the NLo S rays o f gro und reﬂection is d e noted η SB g . T herefor e, the complex c o efﬁcient f o r the NLoS rays of gro und reﬂection can be expr essed as h SB g pq ( t ) = r η SB g Ω + 1 lim N g →∞ N g X n g =1 1 p N g e − j 2 π f c  ξ pn g + ξ qn g  / c × e j 2 π t × f max cos  α ( n g ) R − γ R  cos β ( n g ) R (29) Then, the corresp o nding space CF f o r the NLoS rays o f groun d reﬂection can b e ob tained in a similar metho d above, which is omitted here for brevity . In the mod el, we notice th at the received sign als scattered f rom the grou nd are more likely to be single-bou nced, rather than d ouble-b ounced . Thu s, the space CFs for the single-bo unced r ays of ground reﬂection should be co nsidered. B. Doppler PS D In the V2V cha n nel, the sig n als can propag ate fro m the MT to M R v ia different paths, each of wh ich ca n inv olve reﬂection, diffraction, wa veguidin g, and so on. In add itio n to the ﬂuctuations in the signal envelope and p hase, the received signal frequ ency co nstantly varies as a result o f the relative m otion between the MT and MR. Here, let us deﬁne S ( γ ) as th e Dop pler spec tr um of the p roposed 3D V2V time-variant channel model. In th is case, the receiv ed signals are fo rmed by the sing le-boun ced rays scattered from the scatterers loca ted on th e l th semi-ellipsoid, as well as the doub le-boun ced rays cau sed by the scatter e rs from the combined single cylinder and the l th semi-ellip soid. Mor eover , 9 it is a ssumed th a t the PDFs of the Dopple r frequen cy at the MR are three independ ent random variables; thus, w e can obtain the following characteristic function s as ρ SB l , 3 h l ,pq , h l ,p’q’  t , ω  = Z π − π ρ SB l , 3 h l ,pq , h l ,p’q’  t , ∆ f  e j ω ∆ f d ∆ f (30) ρ SB l , 1 h l ,pq , h l ,p’q’  t , ω  = Z π − π ρ DB l , 1 h l ,pq , h l ,p’q’  t , ∆ f  e j ω ∆ f d ∆ f (31) ρ SB l , 2 h l ,pq , h l ,p’q’  t , ω  = Z π − π ρ DB l , 2 h l ,pq , h l ,p’q’  t , ∆ f  e j ω ∆ f d ∆ f (32) If we take (30 )-(32) into the inverse Fourier transform formu la, the PDF of the total Doppler freq u ency can b e d erived as ρ  t , ∆ f  = 1 2 π Z π − π ρ SB l , 3 h l ,pq , h l ,p’q’  t , ω  ρ SB l , 1 h l ,pq , h l ,p’q’  t , ω  × ρ SB l , 2 h l ,pq , h l ,p’q’  t , ω  e j ω ∆ f d ∆ f (33) Subsequen tly , we deﬁne the Fourier transfo rm of ρ  t , ∆ f  with respect to the variable t . Th en, we can o btain the f unction S ( γ , ∆ f ) as S  γ , ∆ f  = Z π − π ρ  t , ∆ f  e − j 2 π tγ dt (34) If we set ∆ f = 0 , we can then o btain ρ ( t , 0) = ρ ( t ) and S ( γ , 0) = S ( γ ) . There f ore, th e equ ation in (34) can b e rewritten as S  γ  = Z π − π ρ  t  e − j 2 π tγ dt (35) Thus far , th e Dopple r spe ctrum S ( γ ) can be o b tained. Obviously , no te that the proposed Dop pler spectrum does not only de p end on the p roposed chan nel mode l parameters, but also on the n on-stationar y p roperties. I V . N U M E R I C A L R E S U LT S A N D D I S C U S S I O N S In this section, the statistical proper ties of the prop osed 3D non-station ary wideband V2V channel mode l are ev alu ated and analy z e d. The time slots for the stationar y and non- stationary co nditions are set t = 0 an d t = 2 s, respectively . Here, in orde r to invest igate the pr oposed chann el statistics for dif ferent tim e delay s, i.e., per -tap statistics, we deﬁne the semi-majo r dim ensions for the ﬁrst tap a nd second tap are r espectiv ely a 1 = 1 20 m and a 2 = 140 m, i.e., τ = 2( a 2 − a 1 ) / c ≈ 133 ns > 20 ns. Unless o th erwise speciﬁed, all th e chan nel related parameters u sed in this section a re listed in T able I. As mention ed before , th e energy-r elated parameters for tap one an d other taps should be eq ual to un ity , i.e., P 3 i =1 η SB 1 , i + η DB = 1 and η SB l , 3 + η DB l , 1 + η DB l , 2 = 1 . Note th at the e n ergy-related parameter s η SB l , 1 , η SB l , 2 , η SB l , 3 , η DB , η DB l , 1 , an d η DB l , 2 are r elated to the scatter ed cases of NLoS rays, as in [ 3 6]. For example, the received scattered -200 -150 -100 -50 0 50 100 150 200 AoA, α R (degree) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Marginal PDF of AoA P( α R ) α = 20 ° , b 1 = 80m α = 60 ° , b 1 = 80m α = 60 ° , b 1 = 40m α = 180 ° , b 1 = 80m Fig. 5. Margi nal PDF of the AoA statisti cs in the azimuth plane for the dif ferent channel parameter b 1 and diff erent beamwidths of the direction al antenna at the MT . power in tap one high way scen arios is mainly fro m wa ves reﬂected by the stationary road side en vironme n ts. Th e mov- ing vehicles r e presented by the scatterers located o n th e two cylinders are more likely to be single-bo unced, rathe r than double- bounced [21 ,27]. This ind icates that η SB 1 , 3 > max { η SB 1 , 1 , η SB 1 , 2 } > η DB , i.e., η SB 1 , 3 is n ormally larger than 0.4, η SB 1 , 1 and η SB 1 , 2 are normally bo th larger than 0.2 but smaller than 0 .4, while η DB is norma lly smaller than 0.1. For tap on e urban scenarios, the received scattered power is m ainly from the waves scattered from the two-cylind er mod el, i.e ., η DB > max { η SB 1 , 1 , η SB 1 , 2 , η SB 1 , 3 } (no rmally , η DB is larger than 0.6, while η SB 1 , 1 , η SB 1 , 2 , and η SB 1 , 3 are all smaller tha n 0 .15). For tap two highway scenarios, the received scattered power is main ly from waves r e ﬂected by the stationary roadside en vironmen ts de scribed b y the scatterers located on the semi- ellipsoid. Thu s, η SB 2 , 3 > max { η DB 2 , 1 , η DB 2 , 2 } , i.e. , η SB 2 , 3 is normally larger than 0.7, while η DB 2 , 1 and η DB 2 , 2 are both smaller than 0. 15). For tap two urba n scenarios, th e r eceiv ed scattered power is mainly f rom th e double-bo unced rays from the comb ined sing le cylinder and semi-ellipsoid mode ls, i.e., min { η DB 2 , 1 , η DB 2 , 2 } > η SB 2 , 3 (norm a lly , η SB 2 , 3 is smaller than 0.1, while η DB 2 , 1 and η DB 2 , 2 are both larger than 0.4) . On the other hand , the en vironme nt-related param eters k ( l , 1) , k ( l , 2) , and k ( l , 3) are related to the distribution of scatterers. For example, higher v alues of k ( l , 1) and k ( l , 2) (i.e., nor m ally both smaller than 1 0) result in the f ewer moving vehicles/scatterers, i.e., th e hig h way scenarios. In bo th the highway an d ur ban scenarios, k ( l , 3) is large (i.e., norma lly larger than 10 ) as the scatterers reﬂected from roadside environmen ts are no rmally concentr a ted. In addition , Ricean factor Ω is small ( i.e., normally smaller than 1.5 ) in ur ban scenarios, as the LoS compon ent does not h ave d ominant p ower . Howe ver , Ω is large (i.e., nor m ally larger than 3.5) in highway scenarios as fewer moving vehic le s/o b stacles (between the MT an d MR) on the road result in the strong LoS pr opagation componen t. Although the MT and M R in the pro posed model are employed in ULA omn i-directional an tenna e lements, the propo sed model can also be used to a n alyze ra diation p atterns 10 T ABLE I C H A N N E L R E L AT E D PA R A M E T E R S U S E D I N T H E S I M U L AT I O N S T ap one highway scenarios T ap one urban scenarios T ap two highway scenarios T ap two urban scenarios All scenarios D = 200 m, a 1 = 120 m, a 2 = 140 m, f c = 5 . 4 GHz, v R = 54 km/h, ψ T = θ T = π / 3 , ψ R = θ R = π / 3 . Basic parameters R t = R r = 40 m, v R = 25 m/s, f max = 433 Hz R t = R r = 20 m, v R = 8 . 3 m/s, f max = 144 Hz R t = R r = 40 m, v R = 25 m/s, f max = 433 Hz R t = R r = 20 m, v R = 8 . 3 m/s, f max = 144 Hz Rician factor Ω = 3 . 942 Ω = 1 . 062 Ω = 3 . 942 Ω = 1 . 062 Energy-related parameters η SB 1 , 1 = 0 . 371 , η SB 1 , 2 = 0 . 212 , η SB 1 , 3 = 0 . 402 , η DB = 0 . 015 η SB 1 , 1 = η SB 1 , 2 = 0 . 142 , η SB 1 , 3 = 0 . 085 , η DB = 0 . 631 η SB 2 , 3 = 0 . 724 , η DB 2 , 1 = η DB 2 , 2 = 0 . 138 η SB 2 , 3 = 0 . 056 , η DB 2 , 1 = η DB 2 , 2 = 0 . 472 Envir onment-related parameters k (1 , 1) = 8 . 9 , k (1 , 2) = 2 . 7 , k (1 , 3) = 12 . 3 k (1 , 1) = 0 . 55 , k (1 , 2) = 1 . 21 , k (1 , 3) = 12 . 3 k (2 , 1) = 8 . 9 , k (2 , 2) = 2 . 7 , k (2 , 3) = 12 . 3 k (2 , 1) = 0 . 55 , k (2 , 2) = 1 . 21 , k (2 , 3) = 12 . 3 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Normalized antenna spacing δ R / λ Normalized space CF SB 1,1 , ψ T = π /6, θ T = π /6 SB 1,1 , ψ T = π /3, θ T = π /3 SB 1,3 , ψ T = π /6, θ T = π /6 SB 1,3 , ψ T = π /3, θ T = π /3 Reference [28] Fig. 6. Absolut e v alues of time-v ariant space CFs of the single-bounce d models for dif ferent transmit antenna angles in tap one highway scenarios. speciﬁc to the elem ents, which m ake the prop osed geometric channel mo del irregularly shaped. Here , we assume that the transmitter emits the sign al to th e r eceiv er in signiﬁcan tly small beam widths, spa nning th e azimuth ran ge o f [ − α, α ] . It is stated in [ 16] that the AoA statistics of the multi- path compo nents can b e u sed to ev aluate the perf ormance of MIMO comm unication systems. Here, the marginal PDF of the Ao A statistics cor r espondin g to the roa d width b 1 and the b eamwidths of the directional antenna (i.e., α ) at the MT is shown in Fig. 5. It is apparen t that, when the MT is employed with the direction al antenna elements, the AoA PDFs in 0 ≤ α ( n l, 3 ) R ≤ π ﬁrstly decrease to a local value of AoA and the n inc rease to a local maximu m with a “corner” , the AoA PDFs ﬁnally decrease sharp ly , dep ending upon the propo sed geometric chann el model, as seen in Figs. 1 an d 2. A similar behavior can be seen in − π ≤ α ( n l, 3 ) R ≤ 0 . By increasing the beamwidth s α with more scatterers in the scattering region illumin ated by th e directiona l antenna, the PDFs ﬁrstly have higher values on both sides of the curves, and then gradu ally tend to be equal. It can also be no ted that when the road wid th b 1 increases f rom 40 m to 80 m, the 0 0.5 1 1.5 2 2.5 3 3.5 4 Normalized antenna spacing δ R / λ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Normalized space CF Tap 1 , t = 0 (WSS) Tap 1 , t = 2s (non-WSS), γ R = π /3 Tap 1 , t = 2s (non-WSS), γ R = 2 π /3 Tap 2 , t = 0 (WSS) Tap 2 , t = 2s (non-WSS), γ R = π /3 Tap 2 , t = 2s (non-WSS), γ R = 2 π /3 Reference [22] Fig. 7. Absolute values of the time-va riant space CFs of the single-bounced semi-elli psoid m odel for dif ferent taps of the proposed model in highway scenari os. values of the AoA PDFs in c rease sharply . By adop ting an MT antenna element spacing δ T = λ , th e absolute values of the time - variant space CF of the pr o posed V2V chann el mod e l are illu strated in Figs. 6, 7, and 8. By imposing i = 1 and 3 in (23 ) , Fig. 6 shows th e absolute values of tim e-variant space CFs o f the single-bo unced models (i.e., SB 1 , 1 and SB 1 , 3 ) fo r different tra nsmit anten na azimuth angles ψ T and elevation angle θ T . I t is o bvious that the spatial correlation gradu ally decr eases whe n th e normalized antenna spacing d · λ − 1 increases. A similar behavior can be seen in [28]. Add itionally , it is evident that the time-variant space CF decreases slowly as the tr ansmit antenna angles (i.e., ψ T and θ T ) decr e ase. Figs. 7 an d 8 illustrate the absolu te values of the time- variant space CFs for different cha n nel con ditions, i.e., WSS and n on-WSS assump tions. By using (24 ) , the absolute values of the time- variant space CFs of the ﬁrst and secon d taps of the single-bou nced semi-ellipsoid model ( i.e., SB l , 3 ) f or different taps and d ifferent relative moving pro p erties (i.e., t a n d γ R ) are shown in Fig. 7. I n this ﬁgure, the higher correlation in the ﬁrst 11 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Normalized antenna spacing δ R / λ Normalized space CF SB 1,1 , γ R = 0 SB 1,1 , γ R = π DB , γ R = 0 DB , γ R = π SB 1,3 , γ R = 0 SB 1,3 , γ R = π Fig. 8. Absolute val ues of the time-v ariant space CFs of the single- and double-b ounced m odels for dif ferent relati ve moving direct ions in highway scenari os. 0 0.5 1 1.5 2 2.5 3 3.5 4 Normalized antenna spacing δ R / λ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Normalized space CF H t = H r = 10 m , D = 40 m H t = H r = 10 m , D = 60 m H t = H r = 20 m , D = 40 m H t = H r = 20 m , D = 60 m Fig. 9. Absolute values of the time-v ariant space CFs for the s ingle- bounced rays of ground reﬂection for dif ferent ant enna hei ghts (i.e., H t and H r ) and dif ferent distances D between the MT and MR. tap is compared to the seco nd tap b ecause of th e dom in ant Lo S rays, wh ic h is in correspo n dence with the results in [22]. By using (2 5) and impo sing i = 1 an d 3 in (23 ), Fig. 8 illustrates the absolute values o f the time-variant space CFs of the single- (i.e., SB 1 , 1 and SB 1 , 3 ) an d double - bounc e d models (i.e., DB ) of the ﬁrst tap in the WSS co n dition (i.e., t = 0 ). The ﬁgure shows th at the relativ e moving directio n s (i.e., γ R ) have no impact on th e distribution o f the time- variant space CFs wh en the propo sed chan nel model is under the WS S assumption . I t can be observed that the time-variant space CF of the single- bounc e d SB 1 , 3 is lo wer th an that of the single-bou nced SB 1 , 1 . This is due to the fact that hig her ge o metric path lengths result in lower correlation as mention ed in [27]. Howev er , in the propo sed m odel, th e p a th length for SB 1 , 3 is obviously lon ger than the p ath length for SB 1 , 1 . Fig. 9 shows the time-variant space CFs fo r the sing le- bounc e d rays of grou nd reﬂection with respect to the different antenna heigh ts (i.e., H t and H r ) an d different distances D between the MT and MR. Fro m the ﬁgure, we can easily 0 5 10 15 20 Frequency separation, ∆ f [MHz] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Normalized time-varying frequency CF t = 0 (WSS) , DB , γ R = π /3 t = 0 (WSS) , DB , γ R = 2 π /3 t = 2s (non-WSS) , DB , γ R = π /6 t = 2s (non-WSS) , DB , γ R = π /3 t = 2s (non-WSS) , DB 2,2 , γ R = π /6 t = 2s (non-WSS) , DB 2,2 , γ R = π /3 Fig. 10. Absolute value s of the time-v ariant frequenc y CFs of the double- bounced models for dif ferent relati ve moving directio ns and diffe rent time instant s in highway scenarios. notice that when the heights of antenna s moun ted o n the MT and MR incre ase fro m 10 m to 20 m, the space CFs d ecrease slowly , irrespective of the h ighway an d urban environments. Additionally , the space CFs decrease g r adually as the MR gets away fro m the MT . This is mainly d ue to the fact th at higher geometric path lengths result in th e lower co rrelation, as in Fig. 8. For V2V scena r ios, it is im p ortant to analyze the imp act of no n-stationarity , includ ing that of the relative moving di- rections (i.e., γ R ) and moving time instants (i.e., t ), on the statistical properties o f the proposed V2V chan nel model. According ly , by u sing (25 ) and (27), Fig. 10 shows the time-varying fre quency CFs of the d ouble-b ounced models (i.e., DB a n d DB 2 , 2 ) corresp onding to the d ifferent relative moving direction s and different moving tim e instants. It is clearly observed th at, for the doub le - bounc e d DB WSS mo del, regardless of what the relative moving directio ns are (i.e. , γ R = π / 3 or 2 π/ 3 ), the curves of the frequ ency CFs between them tend to be the same , which co nﬁrms the analy sis in Fig. 6. Furtherm ore, it is evident th at w h en the recei ver’ s relativ e movin g d ir ection γ R is π / 3 , the v alue o f the time- variant frequ ency CF is relatively h igher than that at γ R = π / 6 . This is bec a use higher geo metric path lengths result in lower correlation , whereas the path leng th for the p ath length at γ R = π / 3 is obvio usly shorter th an in th e oth er c ases [27]. Th en, we o bserve that the f requency CF of the d ouble- bounc e d DB 2 , 2 is lower than th at of the doub le-boun ced DB in the p roposed non -stationary V2V cha n nel m odel. These results well align with those of pr evious work [12] and thus demonstra te the utility of our mo d el. T o under stand the impact of the chan nel model parameter s and non -stationary prop erties on Doppler PSDs given in (3 5 ) for the theo retical m odel, Fig. 11 shows the no rmalized Doppler PSDs of the proposed V2V chann el model fo r dif- ferent relativ e moving direc tions. I t is observed th a t, f o r the direction of γ R = π / 3 , th e Do ppler PSD of the single-bou n ced SB 1 , 1 is larger than that of the sing le-boun ced SB 1 , 3 because of the h igher fadin g loss cau sed b y th e longer ge o metric 12 −100 −50 0 50 100 −50 −40 −30 −20 −10 0 Normalized Doppler frequency, Hz Normalized Doppler PSD (dB) SB 1,1 , γ R = π /3 SB 1,1 , γ R = 2 π /3 SB 1,2 , γ R = π /3 SB 1,2 , γ R = 2 π /3 SB 1,3 , γ R = π /3 SB 1,3 , γ R = 2 π /3 Fig. 11. Normalized Dopple r PSDs of the proposed V2V chan nel model for dif ferent relati ve mo ving directio ns in highwa y scenarios. -100 -80 -60 -40 -20 0 20 40 60 80 100 Normalized Doppler frequency [Hz] -50 -40 -30 -20 -10 0 Normalized Doppler PSD (dB) Tap 1 , SB 1,3 , γ R = 0 Tap 1 , SB 1,3 , γ R = π /2 Tap 1 , SB 1,3 , γ R = π Tap 2 , SB 2,3 , γ R = 0 Tap 2 , SB 2,3 , γ R = π Avazov Model [2] Clarke Model [14] Fig. 12. Normalized Doppler PSDs of the single-bou nced chann el model for dif ferent taps and differen t relati ve m oving direc tions in highway scenarios. path length . It is also evident that, f o r the waves tha t are single-bou nced at the MR (i.e., SB 1 , 2 ), th e relative movin g direction has no impac t on the distribution of th e Dopp ler PSD. Moreover , th is Doppler distribution tends to be a c o n ventional U-shaped distribution, as shown in [10]. Fig. 12 shows the normalize d Doppler PSDs of the single - bounc e d chan nel mod els (i.e., SB 1 , 3 and S B 2 , 3 ) f or different taps and d ifferent relative moving d ir ections (i.e., γ R ). It is observed that the Dop p ler freq uency gr adually decr eases with a decrease in the taps of the p roposed channe l mo del. It is also apparen t that, for the MR movement p erpendicu lar to the d ir ect LoS rays ( i.e., γ R = π / 2 ), Doppler freq uency in stationary chann el mod el has a similar behavior to tha t o f the re su lts in [ 2] with a p eak at zero. Howev er , this is no t necessary for the p roposed non-station ary V2 V chann el mo d el. W e thus conclud e that the Dopp ler spectru m in non-stationary V2V chann els cha nges con tinually at d ifferent time instants when γ R is set π / 2 , as rep orted in [25]. In addition, if we neglect the elevation angles around th e receiver , the received signal com es fr om the single-b ounced rays (i.e. , SB 1 , 2 ) caused by th e scatterers u niform ly located on a circle ar ound the M R. 1 1.2 1.4 1.6 1.8 2 Time Delay, τ ' (s) 0 0.2 0.4 0.6 0.8 1.0 Normalized Impulse Response Tap 1 , SB 1,1 , R t = 20m (Urban) Tap 1 , SB 1,1 , R t = 40m (Highway) Tap 1 , SB 1,3 Tap 2 , SB 2,3 Fig. 13. Absolut e value of the impulse response of the proposed single- bounced channel m odel for differe nt taps and diffe rent V2V scenarios. Thus, th e p r oposed Doppler PSD is given by the classic Clarke spectrum, which align s with the resu lts in [14]. Meanwhile, Fig. 13 illustrates th e values of the imp ulse response of the propo sed 3 D model for different time delays. In the ﬁgure, time delay τ ′ can be d eﬁned as the r atio of the g eometric path lengths and light veloc ity c . Th e shor test and longe st prop agation delays of the prop o sed WSS model are r e sp ectiv ely o btained as τ ′ min = D / c and τ ′ max ≈ 2 a l / c . Furthermo re, it is evident that the imp ulse r e sponse gradually decreases with an increase in tim e delay τ ′ , which ag rees with the results in [27 ] . In addition, the ch a nnel re sp onse gr adually decreases with an in crease in the taps of the prop osed chann el model, wh ich is in ag reement with the theo retical analy sis in Figs. 7 and 12. It is also apparent th at the lower imp ulse channel is R t = 20 m compared to R t = 40 m because of the faster chann el fading. The analy sis above agre es with the results reported in [21] , which can thus be fully utilized for the future d e sign of wireless com munication systems. V . C O N C L U S I O N In this pap er , we hav e proposed a 3D wideba n d geometry- based chann el model for V2V comm unication scenarios. The relativ e movement b etween the MT and MR results in the time-variant geo metric statistics th at make our model no n - stationary . The p roposed model ad opts a two-cylind er mod el to d epict movin g vehicles (i.e., aroun d the MT or MR), as well as mu ltiple confoca l semi- ellipsoid models to mim ic stationary roadside en vironmen ts. Based on experim ental results, these channel statistics sho w different beh aviors at dif ferent r e lati ve moving time instants, th ereby demonstra tin g the cap ability of th e pr o posed model in d epicting a wide variety of V2V en vironmen ts. It is ad d itionally shown that the do minance of the LoS compon ent results in a h ig her corre lation in the ﬁrst tap of the prop osed chan nel model than in the seco nd one. From the num erical results, we co nclude that the time-variant space CF and f requen cy CF are signiﬁcantly affected b y th e different tap s of th e propo sed time-variant chann el m odel, the relativ e moving times, and the dire c tions between the MT and MR. Finally , it is sho wn that the p roposed model closely 13 agrees with the measure d data, which validates the utility of our mod el. V I . A C K N O W L E D G E M E N T S The au thors would like to th ank Profe ssor Hikmet Sar i, Departmen t of T elecommu nication and In formatio n Engineer- ing, Nanjing University of Posts and T elecom m unications, China, for helping us complete th is study su ccessfully . The authors would also thank the anonymou s revie wers for their constructive comments, which g reatly helped improve this paper . R E F E R E N C E S [1] DOCOMO 5G White Paper , “5G radio access: requiremen ts, concept and technol ogies, ” NTT DOCOMO, INC. , Jul. 2014. [2] N. A vazo v and M. Patz old, “ A geometric street scattering channel m odel for car -to-car communication systems, ” Internati onal Conf. on A dvance d T echnol ogie s for Commun. (ATC 2011) , Da Nang, V ietnam, Aug. 2011 , pp. 224-230. [3] C. X. W ang, X. Cheng, and D. I. Laurenson, “V ehicle-t o-ve hicle channel modeling and m easuremen ts: recent adv ances and future challeng es, ” IEEE Commun. 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Andersen, Channe ls, Propagatio n and Antennas for Mobile Communicat ions, Institution of Electric al Engineers, 2003. 1 Three-Dimensional No n -Stationary W ideband Geometry-Based Ch annel Model for MIMO V ehicle -to-V ehicl e Communication Sy stems Hao Jian g, Zaichen Zhang , Sen ior Member , IEEE , Xiaodon g W a ng, F e llow , IEEE , Liang W u, Memb er , IEEE , and Jian Dang, Member , IEEE Abstract —In this paper , we present a three-dimensional (3D) non-wide-sense stationary (non-WSS) wideband geom etry-based channel model fo r vehicle-to-v ehicle (V2V) communication en- vironments. W e in troduce a two-cylinder model to describe moving vehicles as well as multiple confocal semi-ellipsoid models to d epict stationary roadside scenarios. Th e received signal is constructed as a sum of the line-of-sight (LoS), sin gle-, an d double-boun ced rays wi th different energies. Accordin gly , the proposed channel model i s sufﬁcient for d epicting a variety of V2V scenarios, such as macro-, micro-, and picocells. The relati ve movement b etween t he mobile transmitter (MT) and mobile r eceiv er (MR) results in time-varia nt geometr ic statistics that make our chann el model non-stationary . Usin g this chann el model, the p roposed chann el statistics, i.e., th e time-var iant sp ace correla tion functi ons ( C F s), frequency CFs, and corresponding Doppler power sp ectral density (PS D), were studied for different relati ve mo ving time instants. The nu merical results demonstrate that the p roposed 3D non-WSS wideband chann el model is practical f or characterizing r eal V2V channels. Index T erms —3D channel model, vehicle-to-vehicle communi- cation en vironments, time-variant space and frequency corr ela- tion functions, Doppler power sp ectral density . I . I N T R O D U C T I O N A. Motivation Recently , vehicle-to -vehicle (V2 V) comm unications have received considerab le attention o n accoun t of the rapid de- velopment of ﬁfth- g eneration (5G) wireless comm unication networks [1]. Unlike conv entional ﬁxed-to-m obile (F2 M) cel- lular systems, V2V systems a r e employed with low-ele vation multiple antenn as, and the mob ile tra n smitter (MT) and mobile receiver (MR) are both in relative motion. In V2V scen arios, multiple-inp ut and multiple-ou tput (MI MO) techn o logy is becoming incre asingly attractive becau se large-scale antenna elements can be easily m ounted on vehicular surfaces. For the d evelopment of V2 V commun ication systems, th e radio propag ation ch aracteristics must be d esigned b etween the MT and MR [2-3 ]. Cha n nel mode lin g is often deem ed an effecti ve approa ch f or in vestigating actual p hysical attenuation by reliably d escribing the chann e l characte r istics [4-6]. This work is supported by national key research and de vel opment plan (No. 2016YFB050220 2), NSFC project s (6157110 5, 6150110 9, 61601119 , and 61601120), and scienti ﬁc research foundation of graduate school of Southea st Uni ve rsity (No. YBJJ1655). H. Jiang, Z . Zhang, L. W u, and J. Dang are with the National Mobile Communicat ions Research Laboratory , Southeast Uni versi ty , Nanjing 210096, P . R. China. (e-mail s: { jiangh ao, zczhang, wuliang, ne wwanda } @seu.e du.cn) X. W ang is with the Department of Electrical Engineerin g, Columbia Uni ve rsity , New Y ork City , NY 10027 USA (email: wangx@ee .columbia.e du). B. Prior W ork 1) Geometry-Ba sed Channels T o ev alua te the perfo rmance of a V2V c o mmunica tio n system, accu r ate chan n el models ar e indispe n sable. In ter ms o f the mode ling approach , V2V ch annel models can be ca tego- rized as g e o metry-b ased deterministic models ( GBDMs) [7] , non-g eometry- based stochastic models (NGBSMs) [8], an d geometry -based stochastic mo dels (GBSMs). The latter can be f u rther classiﬁed as regular-shaped GBSMs (RS-GBSMs) [9]-[1 4] o r irr egular-shaped GBSMs ( IS-GBSMs) [1 5]-[1 7 ], depend ing on whether e ffecti ve scatterers are loc ated on regular shapes, e. g., one-r ing, two-ring, ellipses, or irregular shapes. In [9], the auth ors dem onstrated that the line-o f-sight (LoS) is more likely to be ob structed by buildings and obstacles between the MT a n d MR. T hus, it is necessary to develop Rayleigh chan n els to describe the V2V en v ironmen ts. In [9] , an RS-GBSM f or V2V Ray leigh fading chan nels was ﬁrstly introd u ced. Ref. [9 ] was th e ﬁrst to propo se an RS- GBSM for V2V Rayleigh fading channe ls. In additio n, Cheng [11] introduced an RS-GBSM for V2V scenarios. The author presented a two-ring mo del to depict the moving scatterers and multiple con focal ellipses to mimic static scatterers. Y uan [12] adopted a tw o-sphere model to describe m oving vehicles as well as multiple confo cal ellip tic-cylinder mod e ls to depict stationary roadside scenarios. In 2014, Zajic [13 ] pro posed a two-cylinder mo del to d epict movin g and stationary scatterers around the transmitter a n d rec e i ver . Accord in gly , in [ 13], the autho rs stated th at the mob ility o f scatter ers signiﬁcantly affects the Doppler spectr um; therefor e, it is important to cor - rectly acco unt for that effect. Furthermo r e, th ree-dimen sional (3D) RS-GBSMs fo r m acrocell and mic r ocell communication en vironmen ts were respectively pr esented in [15] and [16]. Howe ver, most of th e ab ove RS-GBSMs fo cus on n arrowband channel mo dels, wher ein all rays experience a similar pr opa- gation delay [17] . T his u nrealistically describes wireless co m- munication en vironme n ts. Accord ing to the ch annel measure- ments b etween the n a rrowband and wideb a nd V2V chann els in [18], Sen concluded that the ch a nnel statis tics for different time delay s in wideba nd c hannels should be addr essed. In 2009, the authors in [1 9] ﬁrst proposed a wideband RS-GBSM for MIMO V2V Ricean fading ch annels. Howe ver, in [19], the mo del was shown to be u n able to describe the channel statistics for different time delays, which are signiﬁcan t for 2 wideband chann els. Based on two measured scenarios in [20 ], Cheng [21] introduced the concep t of high vehicle trafﬁc density (VTD) and low VTD to represent movin g veh ic les, respectively . Th at au thor p r esented the chan n el statistics for different time d elays (i.e., per-tap channel statistics); neverthe- less, th e angular sprea d ing of incident waves in a n elevation plane in 3 D space was ignor ed. Most p r evious chann el models r ely o n the wide-sense stationary (WSS) assumption. Ac cording ly , chan n el statistics are assumed to be unchang ed with respect to time. Based on the measurement results in [22], the WSS assumption is valid only for the conventional F2M channe ls fo r very short time intervals, e.g., in the order o f m illiseconds. The development of V2 V channe l measuremen ts has prompted resear chers to re-ev aluate the v alidity of n on-wide- sense statio n ary (n on- WSS) co n ditions. In [23] an d [24] , the authors pro posed two-dimensional (2D) geometry -based n on-WSS n arrowband channel mo dels for T -junction a n d straigh t road environments, respectively . Additionally , [2 5] and [26] pr esented 2 D non- stationary theor etical wideband MIMO Ricean channels for V2V scenario s. Howe ver , these chann els remain restricted to research in a n azimu th plane. Furtherm o re, Y uan [ 27] pre- sented a 3D wideb and MIMO V2V ch annel. Non etheless, that author o n ly focused o n two relative special moving direction s: the sam e d ir ection an d o pposite d ir ection. The authors of [28] presented a wide b and MIMO mod e l for V2V channe ls based on extensive measu r ements taken in highway and rural en vironmen ts. In their study , the effects o f the mob ile d iscr e te scatterers, static discrete scatterers, and diffuse scatterers o n the time-variant channel properties were investigated. For the above-mentione d channel models, they d id not analy ze the mobile pr o perties between the M T and MR, in c lu ding the relativ e moving time and moving d irections [ 5 ]. Howev er , th e time-variant spac e-time and f requen cy CFs, which are impor- tant for V2V chan nels, were not stud ied in d etail. There fore, these models canno t realistically descr ib e V2V commun ication en vironmen ts. C. Main Contributions In th is paper, we present a 3D non-station ary wideb and semi-ellipsoid mo del for MIMO V2V Ricean fading channels. The model is extend ed from Janaswamy’ s 3 D hem ispheroid channel mod el for mac rocellular wireless environmen ts [33]. Compared with the work in [28], the ch a nnel mo del in this paper is cap able of dep icting a wide variety of co mmunicatio n en vironmen ts by adjusting th e model p arameters. Add itionally , our mod el is time -variant because of the relative mo tion between the MT and MR. Conseq uently , we can analyze the propo sed chan nel statistics for mo re moving directio ns, rather than some special moving condition s as m entioned in [28]. Furthermo re, in the pro p osed model, the effect of road wid th on the V2V chann el statistics can be in vestigated. It is im- portant to analyze the proposed c hannel statistics for different taps and different path delays in n o n-stationary cond itions. This model further corrects the unrealistic assumption widely used in current V2 V RS-GBSMs. F or example, th e authors in [5] adop ted the WSS channel to describe the V2V scenario s; the imp act of non-stationar y on V2V chan nel statistics was neglected. It is assumed that the azimuth angle of dep arture (AAoD), ele vation an gle o f departu r e (EAoD) , azimuth an gle of arriv al (AAoA), and elevation a n gle of arrival (E AoA) are indepen d ent of each o ther [ 25]. Th e major co ntributions of this paper are outlined as follows: (1) Based on the two measured scenarios mentioned above in [20 ], we pro pose a 3D non-station ary wideband geometric channel mod el for two different V2 V commu n ication envi- ronmen ts, i.e., highway scenarios and urban scen arios. By adjusting the propo sed channe l param eters, our mod el can describe a variety of V2 V scen arios, such as m acro-, micr o-, and picoce ll scena r ios. (2) W e outlin e th e statistical pr operties of the pro posed V2V channe l model for different taps. Im portant time-variant channel statistics are derived and tho rough ly invest igated. Speciﬁcally , the time-variant space an d frequen cy c orrelation function s (CFs) and co rrespond ing Dop p ler power spectral densities ( PSDs) are de riv ed for V2V scen arios with different relativ e moving directions. (3) Th e impacts of non -stationarity (i.e., relativ e moving time and relativ e moving d irections) on time-variant space and frequen cy CFs are in vestigated in a compar ison with those of th e cor respondin g WSS mod el and measured r esults. The results sh ow that th e prop osed chann el mod e l is an excellent approx imation o f the realistic V2V scenarios. (4) The geo metric pa th length s betwee n the MT an d MR in a 3D semi-ellipsoid V2V chan nel m o del continu e to chan ge because th e tra nsmit az imuth an d ele vation angles constantly vary . W e thus analyze th e pr o posed statistical prop erties for different taps an d different path delays, which is a different approa c h than those pr esented in previous works [12,25,2 7]. The remain d er of this p aper is organized as f ollows. Section II details the p roposed theoretical 3D no n-stationary wideband MIMO V2V ch annel model. I n Section II I, based o n the pro- posed geo metric model, the time-variant spa c e CFs, freq uency CFs, and corr esponding Doppler PSDs ar e derived. Numerical results and d iscussions are pr ovided in Section IV . Finally , ou r conclusion s are presented in Section V . I I . A DA P T I V E G E O M E T RY - BA S E D V 2 V T H E O R E T I C A L C H A N N E L M O D E L In V2V scenarios, the impacts of moving vehicles and roadside en viro n ments o n th e ch annel statistical properties should be add r essed [11,21 ]. Additionally , the relativ e move- ment b etween the MT and MR m a kes the V2V channel time-variant. Howe ver , the previous channe l mo dels have certain lim itatio ns in ter ms o f r ealistically d escribing the V2V commun ication environments. F or example, the models in [10] an d [14 ] rely on the WSS assumption, which imp lies that in the time doma in, the channel fadin g statistics remain in variant over a short period of time. Thus, the above chan nel models cou ld not depict the real V2V environments b ecause of the motion b etween th e MT and MR. Th e autho r s in [15 ] and [19] p r esented the semi-ellipsoid and cylind er mode ls, respectively , to describe the scatterers around the MT and MR. Howe ver, in these stu d ies, the effect o f the r o adside 3 ] [ \ ]  \   W  W  < = x  x   a   a  b  b Fig. 1. Proposed 3D RS-GBSM combining the two-cyli nder model and multiple confocal semi-ellip soid m odels with line-of-si ght (LoS) propagation rays for a wideband MIMO V2V channel. ] [ ]  \  \ 07 05 Fig. 2. Geometric angle s and path lengths of the proposed V2V channel model for s ingle- and double-bounce d pro pagati on rays. en vironmen ts on the chann el statistics was not discussed. In [31] and [32], the autho rs pro posed ellipsoid chann el mo d els to describe the mob ile radio environments. Howe ver, th e m oving vehicles arou nd the MT and M R were not in vestigated in V2V environments. On th e other h and, the au thors in [28] perfor med the chan nel m e a surements only with the MT and MR dri ving in the same directio n, and with th e MT and MR dr i ving in o pposite direction s. Howe ver , the effect of the arbitrary moving dire ctions o n the channel statistics was not in vestigated. Motiv ated by the above drawback s, we have adopted a 3D non- statio n ary wideba n d geome tr ic chann el model in this paper to describe the real V2V comm unication en vironmen ts, as illustrated in Figs. 1 and 2. In the p roposed chann e l mod el, we assume tha t the MT and MR a re loc a ted in the same azimuth plan e . Thus, the model is mainly applicable f o r ﬂat ro ad condition s. Similar assumption can be seen in [25] and [27]. Ho wev er , in reality , the vehicles can be anywhere above, b elow , or on the actual slope, requir ing a more careful analy sis to accurately mo del this V2V pro pagation cond ition. For example, the authors of [35] presented path loss ch a nnel models for sloped- terrain scenarios, in which the g round reﬂection was con sid e red in the V2V ch annels. In this study , the author s introduce d four V2V scena rios: (1) two vehicles are located at opposite ends o f the slope; (2) one vehicle is on the slo pe, an d the other vehicle is beyond th e slope crest; (3 ) one vehicle is on the slop e, an d the o ther is away f r om the slope at the bottom; (4 ) both vehicles are on the slope. Fig s. 1 and 2 illustrate the g eometry of the prop osed V2V channel m odel, which is th e com bination of line-o f-sight (LoS), single-, and double- bounc e d propa g ation r ays. Here, we use a two-cylinder model to depict moving vehicles (i. e ., ar ound the MT or MR). W e emp loy mu ltiple confocal semi-ellipsoid models to mimic station ary ro a dside en v ironmen ts. In general, we note that m o st stru ctures in macr ocell scenario s (e.g. , buildings, highways, ur ban spaces) have straight vertical su rfaces. Th us, we adopt vertical cylinders to model the scatterin g surfaces represented by movin g vehicles [1 3 ,19]. Becau se the he ig hts of th e vehicles and pedestrians are similar to tho se of the transmitter and recei ver , we can assume tha t the scatterers lie on the cylinder model at the MT and M R in the pr oposed 3D space. T o ju stify this assum p tion, correspon ding comparisons are m ade b etween the assump tions o f the m oving vehicles of the 2D circle a n d 3D cylinder m odels. The results show that the p ower le vels of the Dopple r spectrum between these models are insigniﬁcant. Additiona lly , we in troduce th e 3D semi-ellipsoid model because of the following p oints. (1 ) It is stated in [ 5] that the 2 D e llip tical chan n el mo dels with MT and MR located at the foci can d e pict realistic V2V scenarios; h owe ver, they n eglect the transmission signal in the vertical plane. (2) Geo metric path lengths between the MT an d MR in a 3D semi-ellipsoid V2V channel mo del continue to change as the tran smit azimuth and ele vation angles constantly vary . Thus, we can analyze the propo sed statistical proper ties for different p ath delays as the tap is ﬁxed. (3) W e c an further analyz e the chan nel statistics for different path delays in different tap s. This approach is signiﬁcantly different fro m those in previous work s [1 2,27]. T o the best of our knowledge, this is the ﬁrst time that a 3D semi- e llip soid m odel is used to mimic V2V ch annels. As shown in Figs. 1 and 2, supp ose tha t the MT and MR are eq uipped with un iform linear array (UL A ) M T and M R omnidire ctional a n tenna elem e nts. The propo sed mo del is also capab le o f introduc ing o th er MIM O geome tr ic an- tenna sy stem s, such as uniform circular array (U CA), unif o rm rectangu la r array (URA), an d L-shap ed array . The d istan ce between th e cen ters o f the MT and MR cylinder s are denoted as D = 2 f 0 , where f 0 designates the half- length of th e d istance between the two focal points of the ellipse. Let us deﬁne a l , b l , and u l as the semi-major axis of the thre e d im ensions of the l th semi-ellipsoid, where b l = p a 2 l − f 2 0 . It is assumed tha t the radius of the c ylindrical surface arou nd th e MT is den oted as R t 1 ≤ R t ≤ R t 2 . Note that R t 1 and R t 2 correspo n d with the respective ur b an and h ighway scenarios in [20]. Similarly , at the M R, the radius of th e cylindrical surface is denoted as R r 1 ≤ R r ≤ R r 2 . L et An t T p represent the p th ( p = 1 , 2 , ..., M T ) antenna o f the transmit ar ray , and let An t R q represent th e q th ( q = 1 , 2 , ..., M R ) an tenna of the receive arr ay . The spaces between the two adjacent antenna elements at the MT and MR are denoted as δ T and δ R , r e spectiv ely . The orientations of the transmit antenna ar ray in the azimuth plane (relati ve to the x - 4 07 05 B /R6  W C A 07 05 M G E F N U V   a   b  W  W l W  W l W Fig. 3. The ellipse model describing the path geometry (a) ﬁrst tap; (b) other taps. axis) and ele vation p lane (relativ e to the x-y plan e ) are denoted as ψ T and θ T , respectively . Similarly , at the receiver , the orientation s are denoted as ψ R and θ R , respectively . Here, we assume that ther e are N 1 , 1 scatterers (moving vehicles) existing on the cylindrica l surface arou nd the MT , an d the n 1 , 1 th ( n 1 , 1 = 1 , ..., N 1 , 1 ) scatterer is deﬁned as s ( n 1 , 1 ) T . N 1 , 2 effecti ve scatterers likewise exist ar ound the MR lying on the cylind e r model, and the n 1 , 2 th ( n 1 , 2 = 1 , ..., N 1 , 2 ) scatterer is deﬁned as s ( n 1 , 2 ) R . For th e m ultiple con focal semi-ellipsoid m o dels, N l , 3 scatterers lie o n a multiple c o nfocal semi-ellipsoid with th e MT and M R located at the foc i. T he n l , 3 th ( n l , 3 = 1 , ..., N l , 3 ) scatterer is d esignated as s ( n l , 3 ) . Although the propo sed cha nnel model only considers the azimuth and elev atio n angles in the 3D space, it can also be used in polarize d antenna ar r ays [40 ], as the po larization ang les are taken into accoun t in th e model. The pr oposed 3D no n-stationary V2V chan nel model is operated at 5.4 GHz, with a ba n dwidth of 50 MHz. Actually , the pro posed ba n d is capable of depicting a variety of V2V en vironmen ts, such as urb an and highway scenarios. Moreover , it can also be used to estimate some o ther V2 V b and con ditions [37]. Note that the proposed band is close to th e 5.9 GHz V2V band [21]. Howe ver , the difference b etween 5.4 and 5.9 GHz is 9.3% ( = 0.5 /5.4); thu s, the ir pro pagation chann el character- istics d o not change signiﬁcantly . Based on the measurements in [38] and [ 3 9], the path loss expo n ent has a variation of less than 15% over 1 GHz band width and the delay spread has less tha n 1 0% variation over 8GHz b a ndwidth. Her e , we could regard these values as a n unce r tainty of the estimated model p arameters a t 5.4 GH z , wh en th e g oal is to estimate parameter values at 5.9 GHz. In multipath cha n nels, the path length of each wave deter- mines the pr opagation delay and essentially also th e average power of the wa ve at the MR. In [ 21], the auth ors state that the ellipse mode l forms to a certain extent the ph y sical basis for th e m o delling of freq uency-selective channe ls. Theref o re, when the MT and MR are located in th e focus of the ellipse, ev ery wave in the scatterin g region characterized b y th e l th ellipses u n dergoes the same discrete pro pagation delay τ ℓ = τ 0 + ℓτ , ℓ = 0 , 1 , 2 , ..., L − 1 , where τ 0 denotes the propag ation delay o f th e L oS comp onent, τ is an in ﬁnitesimal propag ation d e lay , and L is th e n umber of paths with different propag ation delays. In pa r ticular , the numb er of paths ℓ with different pro pagation delay s exactly cor respond s to the numbe r of delay elem e n ts r equired for the tapp ed-delay -line (TDL) structure of mo delling f requen cy-selective chann els. W e ob- serve that in r eal V2V com munication scenarios with different contributions of single- and d ouble-b ounced rays to the V2V channel statistics, it is necessary to design different taps of the pro posed wideba n d V2V cha n nel model. As mentioned in [36], the tap is strong ly related to the delay resolutio n in V 2 V channels. He r e, let us deﬁne a l as th e semi-m ajor of the l th ellipse in the azimu th plane . Then, for th e n ext time de la y , the semi-major o f the ( l + 1) th ellipse in the azimuth plan e can be derived as a l +1 = a l + cτ / 2 with c = 3 × 1 0 8 m/s. Modelling V2V channels by using a TDL stru c tu re with time-variant co efﬁcients gives a deep insight into the channel statistics in the propo sed model. In Fig . 3(a) , we notice that the received signal for the ﬁrst tap is composed of an inﬁnite number o f d elayed and weigh ted replica s of the transmitted signal in a multipath c hannel, inclu ding direct LoS r ays (i.e., MT → MR), single- bounced r a ys caused b y the scatterers located on e ither of the two cylinders (i.e. , MT → A → M R and MT → B → MR) or o n the ﬁrst semi-ellip soid (i.e., MT → C → MR), and d ouble-b ounced ray s genera ted from the scatterers located o n both cylinders (i.e., MT → U → V → MR). Here, let u s deﬁne the comb ination o f the above cases as the ﬁrst tap. Thus, we c a n analyz e the propo sed channel characteristics for different time delays, i.e., per-tap channel statistics, which is meaningfu l for V2V ch annels. Howev er , for o ther tap s ( l ≥ 1 ), the link is a sup erposition of th e sing le- bounc e d ray s th at ar e p roduc e d on ly fr om the scatterers located on the corr espondin g semi-ellipsoid (i.e. , MT → G → MR), as well as the dou b le-boun ced rays caused by the scatterers fro m the comb ined single cylind er (i.e., MT → E → F → MR) and th e corre sp onding semi-ellipsoid (i.e., MT → M → N → MR), as shown in Fig. 3 (b). In g eneral, the pr oposed V2V channel model can be de- scribed by matr ix H ( t ) =  h pq ( t , τ )  M T × M R of size M T × M R . Therefo re, the complex imp ulse re sp onse betwee n the p th transmit anten na and q th rec ei ve antenna in our model can be expressed as h pq ( t , τ ) = P L ( t ) l =1 ω l h l ,pq ( t ) δ ( τ − τ l ( t )) , wh ere the subscr ip t l r epresents th e tap numb er , h l ,pq ( t ) denotes th e complex tap coe fﬁcient of the Ant T p → Ant R q link, L ( t ) is the total num ber of taps, ω l is the attenu ation factors of the l th tap, and τ l is the cor respond in g prop agation time delays [2 5]. A. Pr opo sed 3D channel model description Based on the ab ove analysis, the comp lex tap coe fﬁcient for the ﬁrst tap o f the Ant T p → Ant R q link at the carrier frequen cy 5 f c can be expressed as [1 3][27 ] h 1 ,pq ( t ) = h LoS 1 ,pq ( t ) + 3 X i =1 h SB 1 , i 1 ,pq ( t ) + h DB 1 ,pq ( t ) (1) with h LoS 1 ,pq ( t ) = r Ω Ω + 1 e − j 2 πf c ξ pq / c + j 2 π t × f max cos  α LoS R − γ R  cos β LoS R (2) h SB 1 , i 1 ,pq ( t ) = r η SB 1 , i Ω + 1 lim N 1 , i →∞ N 1 , i X n 1 , i =1 1 √ N 1 , i e − j 2 π f c ξ pq,n 1 , i / c × e j 2 π t × f max cos  α ( n 1 , i ) R − γ R  cos β ( n 1 , i ) R , i = 1 , 2 , 3 . (3) h DB 1 ,pq ( t ) = r η DB Ω + 1 × lim N 1 , 1 , N 1 , 2 →∞ N 1 , 1 , N 1 , 2 X n 1 , 1 , n 1 , 2 =1 s 1 N 1 , 1 N 1 , 2 × e − j 2 π f c ξ pq,n 1 , 1 ,n 1 , 2 / c × e j 2 π t × f max cos  α ( n 1 , 2 ) R − γ R  cos β ( n 1 , 2 ) R (4) where ξ pq,n 1 , i = ξ pn 1 , i + ξ qn 1 , i and ξ pq,n 1 , 1 ,n 1 , 2 = ξ pn 1 , 1 + ξ n 1 , 1 n 1 , 2 + ξ qn 1 , 2 denote the trav el distance o f the waves through the link Ant T p → s ( n 1 , i ) → Ant R q and Ant T p → s ( n 1 , 1 ) T → s ( n 1 , 2 ) R → An t R q , respectively . Her e, Ω de notes the Rice factor an d f max is the maximum Doppler fr equency with respect to the MR [11]. α LoS R and β LoS R denote the AAoA and EAo A of the LoS path, respectively . For th e NLoS r ays, the symbol α ( n 1 , 1 ) R represents the AAoA of the wav e scatter ed from the effecti ve scatterer s ( n 1 , 1 ) T around the MT , whereas α ( n 1 , 2 ) R represents the AAoA of the wav e scattered fro m the scatterer s ( n 1 , 2 ) R around the MR. Similarly , β ( n 1 , 1 ) R and β ( n 1 , 2 ) R denote the EAoAs o f the waves scattered from the scatter er s ( n 1 , 1 ) T and s ( n 1 , 2 ) R , respecti vely . On the other hand, α ( n 1 , 3 ) R and β ( n 1 , 3 ) R denote the AAoA and EAoA of the waves scattered from th e scatterer s ( n 1 , 3 ) in the semi- ellipsoid model fo r the ﬁrst tap . It is evident that the MT and MR are both movin g, which can be equiv a le n t to a static MT situation with the p rinciples o f relativ e motion . Similar work can be seen in [2] an d [5]. In this case, the MR moves in an arbitrary direction , γ R , with a constant velocity of v R at time in stant t in th e azimuth p lane. Furthemore, energy - related parameters η SB 1 , i and η DB specify the amo u nt that the single- and do uble-bo unced rays r espectiv ely con tribute to the total scattere d power , which can be nor malized to satisfy P 3 i =1 η SB 1 , i + η DB = 1 fo r b revity [11,2 1]. Howe ver , as shown in Fig. 2, for other taps ( l ≥ 1), the com plex tap coefﬁcient of the An t T p → Ant R q link can b e derived as h l ,pq ( t ) = h SB l , 3 l ,pq ( t ) + h DB l , 1 l ,pq ( t ) + h DB l , 2 l ,pq ( t ) (5) with h SB l , 3 l ,pq ( t ) = √ η SB l , 3 lim N l , 3 →∞ N l , 3 X n l , 3 =1 1 √ N l , 3 e − j 2 π f c ξ pq,n l , 3 / c × e j 2 π t × f max cos  α ( n l , 3 ) R − γ R  cos β ( n l , 3 ) R (6) h DB l , 1 l ,pq ( t ) = √ η DB l , 1 lim N 1 , 1 , N l , 3 →∞ N 1 , 1 , N l , 3 X n 1 , 1 , n l , 3 =1 s 1 N 1 , 1 N l , 3 × e − j 2 π f c ξ pq,n 1 , 1 ,n l , 3 / c × e j 2 π t × f max cos  α ( n l , 3 ) R − γ R  cos β ( n l , 3 ) R (7) h DB l , 2 l ,pq ( t ) = √ η DB l , 2 × lim N l , 3 , N 1 , 2 →∞ N l , 3 , N 1 , 2 X n l , 3 , n 1 , 2 =1 s 1 N l , 3 N 1 , 2 × e − j 2 π f c ξ pq,n l , 3 ,n 1 , 2 / c × e j 2 π t × f max cos  α ( n 1 , 2 ) R − γ R  cos β ( n 1 , 2 ) R (8) where ξ pq,n l , 3 = ξ pn l , 3 + ξ qn l , 3 , ξ pq,n 1 , 1 ,n l , 3 = ξ pn 1 , 1 + ξ n 1 , 1 n l , 3 + ξ qn l , 3 , a n d ξ pq,n l , 3 ,n 1 , 2 = ξ pn l , 3 + ξ n l , 3 n 1 , 2 + ξ qn 1 , 2 denote the travel distance o f th e waves throug h the link An t T p → s ( n l , 3 ) → An t R q , Ant T p → s ( n 1 , 1 ) T → s ( n l , 3 ) → Ant R q , and Ant T p → s ( n l , 3 ) → s ( n 1 , 2 ) R → Ant R q , respecti vely . α ( n l , 3 ) R and β ( n l , 3 ) R denote the AAoA and EAoA of the waves scattered from the scatterer s ( n l , 3 ) in th e l th semi-ellipsoid mo del for other tap s. Similar to the ab ove case, en ergy-related param eters η SB l , 3 and η DB l , 1 ( η DB l , 2 ) specify the amount th at th e single- an d do uble-bo unced rays respectively co ntribute to the to tal scattered power , wh ich can be normalized to satisfy η SB l , 3 + η DB l , 1 + η DB l , 2 = 1 for brevity . In addition, because th e deriv atio ns of the cond ition that gu arantees the f ulﬁllment of th e TDL structure are the same, we on ly detail the deriv ation of the condition f or the second tap. As intro duced in [12] and [2 7], we note tha t the impulse response of the propo sed mod e l is relate d to the scatter e d power in V2V chann e ls. T h erefore , it is importan t to d eﬁne the r e ceiv ed scattered power in d ifferent taps and different V2V scenarios (i.e., highway an d urban scenarios) in the propo sed non-stationary channel mo del. In sh ort, fo r the ﬁrst tap, the single-bo unced rays are caused by the scatterers located o n either of the two cylinders or the ﬁrst semi- ellipsoid, while th e dou b le-boun ced ra y s are gener ated from the scatterers located on th e b oth cylinder s, as shown in Fig. 2. For highway scenar ios (i.e., R t = R t 2 and R r = R r 2 ), the higher relative movement of the vehicles results in a h igher Doppler frequen cy; moreover , the value of Ω is always large because the LoS compon ent can bear a signiﬁcant amou nt of power . Additionally , the r eceiv ed scattered power is mainly from wa ves reﬂected by the stationary roadside en vir o nments described by the scatterer s located on the ﬁr st semi-ellipsoid. The moving vehicle s represented by the scatterers located on the two cylinders are mo re likely to be single- b ounced , rather th a n doub le-boun ced. T his ind icates that η SB 1 , 3 > max { η SB 1 , 1 , η SB 1 , 2 } > η DB . For urban scenarios (i.e., R t = R t 1 and R r = R r 1 ), the lower relative movemen t of th e vehicles results in a lower Doppler frequency; mor eover , the v alue of Ω is smaller th a n that in the h ighway scenarios. Addition a lly , the double- bounc e d ray s of th e two-cylinde r mod el can bear mor e energy than the single-bou nced r ays of th e two-cylinder a nd semi-ellipsoid models, i.e., η DB > max { η SB 1 , 1 , η SB 1 , 2 , η SB 1 , 3 } . 6 \ [ 07 05 05 Fig. 4. T op view of the geometric angles and path lengths in the proposed non-stati onary V2V channel model. Howe ver, for the secon d tap, it is assumed tha t the single- bounc e d r ays are produced only from the static scatter ers located on th e corre sponding sem i-ellipsoid, while the doub le- bounc e d rays are caused by the scatterer s from the combined one cylinder (eith er o f th e two cylinders) and the correspond- ing semi-e llip soid [21 ,26,27 ]. Note that, in th e p r oposed TDL structure, th e double- bounc e d rays in the ﬁrst tap must be smaller in distan ce than the single-b ounced r ays on the second semi-ellipsoid, i.e., max { R t , R r } < min { a 2 − a 1 } . It is stated in [ 41] that the dela y resolution is app roximately the inverse of bandwidth and therefore, we a ssum e that th e delay resolution in the pro posed m odel is 2 0 ns for 50 MHz. In this pap er , we deﬁne d ifferent time d e lays with th e different ellipses. Thus, the second ellipse should produ ce at least 6 m excess path length than the ﬁrst ellipse, i.e., 2 a 2 − 2 a 1 = c τ with τ = 20 ns. In this case, the prop o sed chan nel statistics for different time delays, i.e., per-tap statistics, can be inv estigated. For hig hway scenario s, the received scattered p ower is mainly from wa ves reﬂected by the stationary roadside en vironme n ts described by the scatter ers located on th e semi-ellipsoid, i.e. , η SB l , 3 > max { η DB l , 1 , η DB l , 2 } . For urban scen arios, the do uble- bounc e d rays from the combined sin g le cylinder and semi- ellipsoid mode ls can bear more en ergy than the single-boun ced rays o f the semi-ellip so id mo del, i.e., min { η DB l , 1 , η DB l , 2 } > η SB l , 3 . B. Non-station ary time-variant parameters T o describe the non- stationarity o f the pr o posed 3D wide- band cha n nel model, we introd uce a V2V c o mmunica tion sce- nario, as illustrated in Fig. 4. T he ﬁgure sh ows the geom etric proper ties and moving statistics of the proposed model in the azimuth p la n e. In this ca se, owing to overly com plex issues, the cor respondin g 3 D ﬁgure with MIM O antennas is omitted for brevity . For V2V scenario s, the geom e tric paths len g ths will be time-variant becau se of the relative movement between the MT and M R. Co n sequently , ξ pq , ξ pn 1 , 2 , ξ qn 1 , 1 , ξ qn l , 3 , an d ξ n 1 , 1 n 1 , 2 can b e rep laced by ξ pq ( t ) , ξ pn 1 , 2 ( t ) , ξ qn 1 , 1 ( t ) , ξ qn l , 3 ( t ) , and ξ n 1 , 1 n 1 , 2 ( t ) , resp e c ti vely . Howe ver , in Fig. 4, no te that th e distances ξ pn 1 , 1 , ξ pn l , 3 , and ξ qn 1 , 2 have no related to the non- stationary properties, i.e., ξ pn 1 , 1 ( t ) = ξ pn 1 , 1 , ξ pn l , 3 ( t ) = ξ pn l , 3 , and ξ qn 1 , 2 ( t ) = ξ qn 1 , 2 . In g eneral, it is clear ly observed that the MR is relatively far f r om th e MT in th e pr oposed V2V co mmunicatio n e nvironmen ts. Thus, we can make the following assumptions: min { R t , R r , u − f } ≫ max { δ T , δ R } , D ≫ max { δ T , δ R } , and the appro ximation √ x + 1 ≈ 1 + x / 2 is used for small x . Accord ingly , b a sed o n the law of cosines in appropr iate triang les and small ang le appr oximation s (i.e., sin x ≈ x and cos x ≈ 1 for small x ) [12,2 7], the corr espondin g time-variant geometr ic path lengths at r elati ve moving time instant t can be a pprox im ated as ξ pq ( t ) ≈ q  D − δ Tx  2 +  v R t  2 − 2  D − δ Tx  v R t cos  α LoS R − γ R  (9) ξ pn 1 , 1 ( t ) ≈ R t − h δ Tx cos α ( n 1 , 1 ) T cos β ( n 1 , 1 ) T + δ T y sin α ( n 1 , 1 ) T cos β ( n 1 , 1 ) T + δ Tz sin β ( n 1 , 1 ) T i (10) ξ pn 1 , 2 ( t ) ≈   D − Q p cos θ T  2 +  v R t  2 − 2  D − Q p cos θ T  v R t cos  α ( n 1 , 2 ) R − γ R   (11) ξ pn l , 3 ( t ) ≈ 2 a 2 l b 2 l u 2 l ξ l, 3 − δ T cos θ T h R r / D sin ψ T sin α ( n l , 3 ) T + cos ψ T i (12) ξ qn 1 , 2 ( t ) ≈ R r − h δ Rx cos α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R + δ Ry sin α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R + δ Rz sin β ( n 1 , 2 ) R i (13) ξ qn 1 , 1 ( t ) ≈   D − Q q cos θ R  2 +  v R t  2 − 2  D − Q q cos θ R  v R t cos  α ( n 1 , 1 ) R − γ R   (14) ξ qn l , 3 ( t ) ≈  ξ 2 pn l , 3 ( t ) sin 2 β ( n l , 3 ) T + ξ 2 R +  v R t  2 + 2 D ξ R cos  γ R + α ( n l , 3 ) R   (15) ξ n 1 , 1 n 1 , 2 ( t ) ≈ q D 2 +  v R t  2 − 2 Dv R t cos  α LoS R − γ R  (16) where δ Tx = δ T cos θ T cos ψ T , δ T y = δ T cos θ T sin ψ T , δ Tz = δ T sin θ T , δ Rx = δ R cos θ R cos ψ R , δ Ry = δ R cos θ R sin ψ R , δ Rz = δ R sin θ R , Q p = δ T cos θ T  R r / D sin ψ T sin α ( n 1 , 2 ) R + cos ψ T  , ξ l, 3 = b 2 l u 2 l cos 2 β ( n l , 3 ) T cos 2 α ( n l , 3 ) T + a 2 l u 2 l cos 2 β ( n l , 3 ) T sin 2 α ( n l , 3 ) T + a 2 l b 2 l sin 2 β ( n l , 3 ) T , ξ R =  D 2 + ξ 2 pn l , 3 ( t ) cos 2 α ( n l , 3 ) T − 2 D ξ pn l , 3 ( t ) cos β ( n l , 3 ) T cos α ( n l , 3 ) T  , and Q q = δ R cos θ R  R t / D sin ψ R sin α ( n 1 , 1 ) T − cos ψ R  . T o jointly con sid er th e impact of the azimu th and elevation angles on chann el statisti cs, several scatterer d istributions, 7 such as uniform , Gaussian, Laplacian, and von Mises, were used in pr ior w ork. Here, we adopt the v o n M ises pro bability density f unction (PDF) to ch a racterize the distribution of scat- terers in the pro posed V2V c h annel be c ause it approx imates many o f th e previously men tioned d istributions and leads to closed-for m solution s for many useful situations. Therefore, the von Mises PDF is derived as p ( α ( n l , i ) R , β ( n l , i ) R ) = k cos ( n l , i ) R 4 π sinh k × e k cos β 0 cos β ( n l , i ) R cos  α ( n l , i ) R − α 0  × e k sin β 0 sin β ( n l , i ) R (17) with α ( n l , i ) R and β ( n l , i ) R ∈ [ − π , π ) , α 0 ∈ [ − π , π ) . In add ition, β 0 ∈ [ − π , π ) den otes the m e a n values of the azimuth angle α ( n l , i ) R and elev atio n angle β ( n l , i ) R at the receiver , respec ti vely . In addition, k ( k ≥ 0) is a r eal-valued param eter that con trols the concentratio n of the distribution identiﬁed by the mean direction, α 0 and β 0 [12]. As men tioned in Sectio n I, the proposed 3D ch annel mode l can depict a wide variety of commun ication en viro nments b y adjusting the g eometric model p arameters. For example, it is apparen t that when we d o not take the roadside en viro nments into account, the pro posed mo del tends to the Zajic mod el [13,19 ]. Howev er , the proposed mod el can d escribe the previ- ous 3D stationary semi-ellip soid channels as t = 0 , as shown in [ 31] and [32] . In this case, our channel can be d egenerated into a 2D ellip tical channel as mod el parameter u l is equal to zero. On the oth er h and, wh en we set t 6 = 0 , our mode l can be tra n sformed in to non -stationary V2V chan nels, such as the Ghazal mo del [26] and Y uan mo del [27]. Likewise, the propo sed mod el describes othe r models in p r evious work; we omit them for b revity . I I I . P R O P O S E D T H E O R E T I C A L M O D E L S T A T I S T I C A L P R O P E RT I E S A. Space CFs In genera l, CF is an importan t statistic in d esigning com- munication link that ch aracterizes how fast a wireless channe l changes with respect to time, movement, or frequen cy [25]. The speciﬁc CF that is o f inter e st in this paper is the space CF , which m e a sures th e spatial statistics of the proposed V2V channel. It is stated in [27] th at the spatial correlation p r op- erties o f two arbitra r y channel imp ulse responses h pq ( t ) and h p’q’ ( t ) of a MI MO V 2 V channel ar e co mpletely dete r mined by the cor relation pro perties of h l, pq ( t ) and h l, p’q’ ( t ) in each tap, so th a t no co rrelations exist between the u nderlyin g pr ocesses in different taps. Ther efore, the normalized time-variant ST CF can be expr essed as [36] ρ h l ,pq , h l ,p’q’  t , τ  = E h h l ,pq  t  h ∗ l ,p’q’  t + τ  i r E h   h l ,pq ( t )   2 i E h   h l ,p’q’ ( t + τ )   2 i (18) where ( · ) ∗ denotes the complex co njugate oper ation and E [ · ] is the expectation operation. Because the Lo S, single-, and d ouble-b ounced ray s are independ ent o f each o ther, the channel respon se fo r the ﬁrst tap c a n be expressed as ρ h 1 ,pq , h 1 ,p’q’  t , τ  = ρ LoS h 1 ,pq , h 1 ,p’q’  t , τ  + 3 X i =1 ρ SB 1 , i h 1 ,pq , h 1 ,p’q’  t , τ  + ρ DB h 1 ,pq , h 1 ,p’q’  t , τ  (19) Howe ver, for other taps, accord ing to ( 1 8), we have the time-variant space CF as ρ h l ,pq , h l ,p’q’  t , τ  = ρ SB l , 3 h l ,pq , h l ,p’q’  t , τ  + ρ DB l , 1 h l ,pq , h l ,p’q’  t , τ  + ρ DB l , 2 h l ,pq , h l ,pq  t , τ  (20) By applyin g th e correspond ing scatterer non- unifor m dis- tribution, and by following similar reasoning in [19,2 1], we can o btain the time - variant space CFs o f the Lo S, single - , and double-b ounced rays, as outlined below . Sp e ciﬁcally , by submitting (2) into (18), the time-variant space CF in the case of the Lo S rays can be expr essed as ρ LoS h 1 ,pq , h 1 ,p’q’  t , τ  = K e j 2 πf c c r  D − δ Tx  2 +  v R t  2 − 2  D − δ Tx  v R t cos  α LoS R − γ R  × e j 2 πf max τ cos  α LoS R − γ R  (21) where λ denotes the wavelength. In submitting (3 ) in to ( 18), the time-variant space CF in the case of th e sing le-boun ced rays SB 1 , i can be expressed as ρ SB 1 , i h 1 ,pq , h 1 ,p’q’  t , τ  = η SB 1 , i lim N 1 , i →∞ N 1 , i X n 1 , i =1 1 N 1 , i e j 2 πf c c A ( SB 1 , i ) + B ( SB 1 , i ) (22) where A ( SB 1 , 1 ) =  ( D − Q q cos θ R ) 2 + ( v R t ) 2 − 2( D − Q q cos θ R ) × v R t co s( α ( n 1 , 1 ) R − γ R )  − δ Tx cos α ( n 1 , 1 ) T cos β ( n 1 , 1 ) T − δ T y × sin α ( n 1 , 1 ) T cos β ( n 1 , 1 ) T − δ Tz sin β ( n 1 , 1 ) T , B ( SB 1 , 1 ) = j 2 πf max τ co s( α ( n 1 , 1 ) R − γ R ) cos β ( n 1 , 1 ) R , A ( SB 1 , 2 ) =  ( D − Q p cos θ T ) 2 + ( v R t ) 2 − 2  D − Q p  × v R t co s( α ( n 1 , 2 ) R − γ R )  1 / 2 − δ Rx cos α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R − δ Ry sin α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R − δ Rz sin β ( n 1 , 2 ) R , and B ( SB 1 , 2 ) = j 2 π f max τ co s( α ( n 1 , 2 ) R − γ R ) × cos β ( n 1 , 2 ) R . It is stated in [6] that P N 1 , i n 1 , i =1 1 / N 1 , i = 1 as N 1 , i → ∞ . Thus, the to tal power o f the SB 1 , i rays is proportion a l to 1 / N 1 , i . This is equal to the inﬁnitesimal power com ing fr om the differential of the 3D ang les, d α ( n 1 , i ) R d β ( n 1 , i ) R , i.e., 1 / N 1 , i = p ( α ( n 1 , i ) R , β ( n 1 , i ) R ) d α ( n 1 , i ) R d β ( n 1 , i ) R , wher e p ( α ( n 1 , i ) R , β ( n 1 , i ) R ) d e- notes the joint von Mises PDF in (17). Th erefore , (22) can be rewritten as ρ SB 1 , i h 1 ,pq , h 1 ,p’q’  t , τ  = η SB 1 , i Z π − π Z π − π e j 2 πf c c A ( SB 1 , i ) + B ( SB 1 , i ) × p  α ( n 1 , i ) R , β ( n 1 , i ) R  d α ( n 1 , i ) R d β ( n 1 , i ) R (23) Similarly , submitting (4) into (18) , the time-variant space CF in the case of the single-bounce d rays SB l , 3 can be expressed as 8 ρ SB l , 3 h l ,pq , h l ,p’q’  t , τ  = η SB l , 3 Z π − π Z π − π e j 2 πf c c A ( SB l , 3 ) + B ( SB l , 3 ) × p  α ( n l , 3 ) R , β ( n l , 3 ) R  d α ( n l , 3 ) R d β ( n l , 3 ) R (24) where A ( SB l , 3 ) =  ξ 2 pn l , 3 ( t ) sin 2 β ( n l , 3 ) T + ξ 2 R + ( v R t ) 2 + 2 D × ξ R cos( γ R + α ( n l , 3 ) R )  − δ Tx cos θ T  R r / D sin ψ T sin α T + co s ψ T  and B ( SB l , 3 ) = j 2 π τ cos( α ( n l , 3 ) R − γ R ) cos β ( n l , 3 ) R . Submitting ( 6) in to (1 8), the time- variant spac e CF in the case of the double- bounc e d rays DB can be expressed as ρ DB h 1 ,pq , h 1 ,p’q’  t , τ  = η DB Z π − π Z π − π e j 2 πf c c A ( DB ) + B ( DB ) × p  α ( n 1 , 2 ) R , β ( n 1 , 2 ) R  d α ( n 1 , 2 ) R d β ( n 1 , 2 ) R (25) where A ( DB 1 , 2 ) = δ Tx cos α ( n 1 , 1 ) T cos β ( n 1 , 1 ) T + δ T y sin α ( n 1 , 1 ) T × cos β ( n 1 , 1 ) T + δ Tz sin β ( n 1 , 1 ) T + δ Rx cos α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R + δ Ry sin α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R + δ Rz sin β ( n 1 , 2 ) R and B ( DB 1 , 2 ) = j 2 πτ × cos( α ( n 1 , 2 ) R − γ R ) cos β ( n 1 , 2 ) R . Submitting ( 7) in to (1 8), the time- variant spac e CF in the case of the double- bounc e d rays DB l , 1 can be expr essed as ρ DB l , 1 h l ,pq , h l ,p’q’  t , τ  = η DB l , 1 Z π − π Z π − π e j 2 πf c c A ( DB l , 1 ) + B ( DB l , 1 ) × p  α ( n l , 3 ) R , β ( n l , 3 ) R  d α ( n l , 3 ) R d β ( n l , 3 ) R (26) where A ( DB l , 1 ) =  ξ 2 pn l , 3 ( t ) sin 2 β ( n l , 3 ) T + ξ 2 R + ( v R t ) 2 + 2 D × ξ R cos( γ R + α ( n l , 3 ) R )  − δ Tx cos α ( n 1 , 1 ) T cos β ( n 1 , 1 ) T − δ T y × sin α ( n 1 , 1 ) T cos β ( n 1 , 1 ) T − δ Tz sin β ( n 1 , 1 ) T and B ( DB l , 1 ) = j 2 π × f max τ cos( α ( n l , 3 ) R − γ R ) cos β ( n l , 3 ) R . Submitting ( 8) in to (1 8), the time- variant spac e CF in the case of the double- bounc e d rays DB l , 2 can be expr essed as ρ DB l , 2 h l ,pq , h l ,p’q’  t , τ  = η DB l , 2 Z π − π Z π − π e j 2 πf c c A ( DB l , 2 ) + B ( DB l , 2 ) × p  α ( n 1 , 2 ) R , β ( n 1 , 2 ) R  d α ( n 1 , 2 ) R d β ( n 1 , 2 ) R (27) where A ( DB l , 2 ) = δ Tx R r / D sin ψ T sin α ( n l , 3 ) T cos θ T + δ Tx cos θ T × cos ψ T + δ Rx cos α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R + δ Ry sin α ( n 1 , 2 ) R cos β ( n 1 , 2 ) R + δ Rz sin β ( n 1 , 2 ) R and B ( DB l , 2 ) = j 2 π f max τ co s( α ( n 1 , 2 ) R − γ R ) × cos β ( n 1 , 2 ) R . From (21 )-(27 ), we no tice that th e time-variant space CFs are r e la te d not only to the ge ometric path leng ths, but a lso to the m oving prop erties. By substituting (17) into (21 )-(27 ) , the time-variant space cro ss-functions f or the LoS, single-, and dou ble-bou nced compon ents can be respectively obta in ed. Furthermo re, by setting p = p’ and q = q’ , the time- variant space auto-correlation f unction (A CF) can b e obtained [25]. On the other h and, we note that all above investigated statistical pro perties are time-variant on acco unt of th e non - WSS assump tion of the p r oposed V2V chann el m odel. Con- sequently , b y applying the Fourier transfo rmation o f h l ,pq ( t ) , the time-variant f requency cross-co rrelation f u nction of the propo sed 3 D non - stationary channel model can be d e r iv ed as ρ h l ,pq , h l ,p’q’ ( t , ∆ f ) = E h R ∞ −∞ h l ,pq ( t , τ ) h ∗ l ,pq ( t , τ ) e j 2 π ∆ f τ d τ i r E h   h l ,pq ( t , f )   2 i E h   h l ,p’q’ ( t , f + ∆ f )   2 i (28) Similar to the pr evious case, we substitute the co rrespond ing channel re sponse into (28), an d the time-variant f requen cy CFs for the LoS, single-, and do uble-bo unced compo nents can be r e spectiv ely obtained . Nevertheless, the p roposed channel model under the WSS assump tion (i.e., t = 0 ) demonstra te s that the channel statistics are not dependent on time t . In th is case, th e prop osed channe l model tends to be a conventional F2M channel m o del. Note that the above an alysis is mainly for the ﬂat com- munication environments, where the MR is far from the MT . Howe ver, when the MR is close to the MT , it is impo rtant to in vestigate the effect of groun d reﬂection on th e V2V ch annel statistics [35 ]. Here, we assume that th ere are N g effecti ve scatterers unifo r mly existing on the grou nd in the azimuth plane. The heigh ts o f a n tennas mounte d on th e MT and MR are den oted as H t and H r , respectively . The AAo A an d EAoA of the waves scattered from the scatterer on the grou n d are denoted as α ( n g ) R and β ( n g ) R , r espectiv ely . The distances from the MT and MR to the scatter er o n the groun d are deno ted as ξ pn g and ξ qn g , respectively . The ene rgy-related par ameter for the NLo S rays o f gro und reﬂection is d e noted η SB g . T herefor e, the complex c o efﬁcient f o r the NLoS rays of gro und reﬂection can be expr essed as h SB g pq ( t ) = r η SB g Ω + 1 lim N g →∞ N g X n g =1 1 p N g e − j 2 π f c  ξ pn g + ξ qn g  / c × e j 2 π t × f max cos  α ( n g ) R − γ R  cos β ( n g ) R (29) Then, the corresp o nding space CF f o r the NLoS rays o f groun d reﬂection can b e ob tained in a similar metho d above, which is omitted here for brevity . In the mod el, we notice th at the received sign als scattered f rom the grou nd are more likely to be single-bou nced, rather than d ouble-b ounced . Thu s, the space CFs for the single-bo unced r ays of ground reﬂection should be co nsidered. B. Doppler PS D In th e V2V channel, the receiv ed waves arrive at the MR from v arious d irections (i. e . , NL o S co mponen ts) with different time d elays through multiple paths. In addition to the ﬂu c tu- ations in the signal envelope and phase, the received signal frequen cy constantly varies as a r esult o f the relative motion between the MT and MR. Theref ore, in th e prop osed V2V channel model, the received sign al at the MR incu r s a spre a d in the fre q uency spectr um caused by the relative motion between the MT and MR. Her e, let us deﬁn e S ( γ ) as the Dop pler spectrum of the p roposed 3D V2V time-variant cha n nel model. In this case, the received signal is forme d b y the single- bounc e d rays scattered from the scatterer s located on the l th 9 semi-ellipsoid, as we ll as the do uble-bo unced ray s caused by the scatterers fro m the combin ed single cylinder and the l th semi-ellipsoid. Moreover, the PDFs of the Dop pler frequency at the MR are assumed to be three indepen dent rand o m variables. T o ob tain the PDF of th e total Doppler freq u ency , the char acteristic functions can be deﬁn ed as follows: ρ SB l , 3 h l ,pq , h l ,p’q’  t , ω  = Z π − π ρ SB l , 3 h l ,pq , h l ,p’q’  t , ∆ f  e j ω ∆ f d ∆ f (30) ρ SB l , 1 h l ,pq , h l ,p’q’  t , ω  = Z π − π ρ DB l , 1 h l ,pq , h l ,p’q’  t , ∆ f  e j ω ∆ f d ∆ f (31) ρ SB l , 2 h l ,pq , h l ,p’q’  t , ω  = Z π − π ρ DB l , 2 h l ,pq , h l ,p’q’  t , ∆ f  e j ω ∆ f d ∆ f (32) If we take (30 )-(32) into the inverse Fourier transform formu la, the PDF of the total Doppler freq u ency can b e d erived as ρ  t , ∆ f  = 1 2 π Z π − π ρ SB l , 3 h l ,pq , h l ,p’q’  t , ω  ρ SB l , 1 h l ,pq , h l ,p’q’  t , ω  × ρ SB l , 2 h l ,pq , h l ,p’q’  t , ω  e j ω ∆ f d ∆ f (33) Subsequen tly , we deﬁne the Fourier transfo rm of ρ  t , ∆ f  with respect to the v ariable t to b e th e func tio n S ( γ , ∆ f ) , i.e., S  γ , ∆ f  = Z π − π ρ  t , ∆ f  e − j 2 π tγ dt (34) If we set ∆ f = 0 , we can then o btain ρ ( t , 0) = ρ ( t ) and S ( γ , 0) = S ( γ ) . There f ore, th e equ ation in (34) can b e rewritten as S  γ  = Z π − π ρ  t  e − j 2 π tγ dt (35) Thus far , th e Dopple r spe ctrum S ( γ ) can be o b tained. Obviously , no te that the proposed Dop pler spectrum does not only de p end on the p roposed chan nel mode l parameters, but also on the no n-stationary p roperties, includin g that of th e relativ e time and relative moving directions. I V . N U M E R I C A L R E S U LT S A N D D I S C U S S I O N S In this section, the statistical proper ties of the prop osed 3D non-station ary wideband V2V channel mode l are ev alu ated and analy z e d. The time slots for the stationar y and non- stationary co nditions are set t = 0 an d t = 2 s, respectively . Here, in orde r to invest igate the pr oposed chann el statistics for dif ferent tim e delay s, i.e., per -tap statistics, we deﬁne the semi-majo r dim ensions for the ﬁrst tap a nd second tap are r espectiv ely a 1 = 1 20 m and a 2 = 140 m, i.e., τ = 2( a 2 − a 1 ) / c ≈ 133 ns > 20 ns. Unless o th erwise speciﬁed, all th e chan nel related parameters u sed in this section a re listed in T able I. As mention ed before , th e energy-r elated parameters for tap one an d other taps should be eq ual to un ity , i.e., P 3 i =1 η SB 1 , i + η DB = 1 and η SB l , 3 + η DB l , 1 + η DB l , 2 = 1 . -200 -150 -100 -50 0 50 100 150 200 AoA, α R (degree) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Marginal PDF of AoA P( α R ) α = 20 ° , b 1 = 80m α = 60 ° , b 1 = 80m α = 60 ° , b 1 = 40m α = 180 ° , b 1 = 80m Fig. 5. Margi nal PDF of the AoA statisti cs in the azimuth plane for the dif ferent channel parameter b 1 and diff erent beamwidths of the direction al antenna at the MT . Note th at the e n ergy-related parameters η SB l , 1 , η SB l , 2 , η SB l , 3 , η DB , η DB l , 1 , and η DB l , 2 are r elated to the scatter ed cases of NLoS rays, as in [ 3 6]. For example, the received scattered power in tap one high way scen arios is mainly fro m wa ves reﬂected by the stationary road side en vironme n ts. Th e mov- ing vehicles r e presented by the scatterers located o n th e two cylinders are more likely to be single-bo unced, rathe r than double- bounced [21 ,27]. This ind icates that η SB 1 , 3 > max { η SB 1 , 1 , η SB 1 , 2 } > η DB , i.e., η SB 1 , 3 is n ormally larger than 0.4, η SB 1 , 1 and η SB 1 , 2 are normally bo th larger than 0.2 but smaller than 0 .4, while η DB is norma lly smaller than 0.1. For tap on e urban scenarios, the received scattered power is m ainly from the waves scattered from the two-cylind er mod el, i.e ., η DB > max { η SB 1 , 1 , η SB 1 , 2 , η SB 1 , 3 } (no rmally , η DB is larger than 0.6, while η SB 1 , 1 , η SB 1 , 2 , and η SB 1 , 3 are all smaller tha n 0 .15). For tap two highway scenarios, the received scattered power is main ly from waves r e ﬂected by the stationary roadside en vironmen ts de scribed b y the scatterers located on the semi- ellipsoid. Thu s, η SB 2 , 3 > max { η DB 2 , 1 , η DB 2 , 2 } , i.e. , η SB 2 , 3 is normally larger than 0.7, while η DB 2 , 1 and η DB 2 , 2 are both smaller than 0. 15). For tap two urba n scenarios, th e r eceiv ed scattered power is mainly f rom th e double-bo unced rays from the comb ined sing le cylinder and semi-ellipsoid mode ls, i.e., min { η DB 2 , 1 , η DB 2 , 2 } > η SB 2 , 3 (norm a lly , η SB 2 , 3 is smaller than 0.1, while η DB 2 , 1 and η DB 2 , 2 are both larger than 0.4) . On the other hand , the en vironme nt-related param eters k ( l , 1) , k ( l , 2) , and k ( l , 3) are related to the distribution of scatterers. For example, higher v alues of k ( l , 1) and k ( l , 2) (i.e., nor m ally both smaller than 1 0) result in the f ewer moving vehicles/scatterers, i.e., th e hig h way scenarios. In bo th the highway an d ur ban scenarios, k ( l , 3) is large (i.e., norma lly larger than 10 ) as the scatterers reﬂected from roadside environmen ts are no rmally concentr a ted. In addition , Ricean factor Ω is small ( i.e., normally smaller than 1.5 ) in ur ban scenarios, as the LoS compon ent does not h ave d ominant p ower . Howe ver , Ω is large (i.e., nor m ally larger than 3.5) in highway scenarios as fewer moving vehic le s/o b stacles (between the MT an d MR) on the road result in the strong LoS pr opagation componen t. 10 T ABLE I C H A N N E L R E L AT E D PA R A M E T E R S U S E D I N T H E S I M U L AT I O N S T ap one highway scenarios T ap one urban scenarios T ap two highway scenarios T ap two urban scenarios All scenarios D = 200 m, a 1 = 120 m, a 2 = 140 m, f c = 5 . 4 GHz, v R = 54 km/h, ψ T = θ T = π / 3 , ψ R = θ R = π / 3 . Basic parameters R t = R r = 40 m, v R = 25 m/s, f max = 433 Hz R t = R r = 20 m, v R = 8 . 3 m/s, f max = 144 Hz R t = R r = 40 m, v R = 25 m/s, f max = 433 Hz R t = R r = 20 m, v R = 8 . 3 m/s, f max = 144 Hz Rician factor Ω = 3 . 942 Ω = 1 . 062 Ω = 3 . 942 Ω = 1 . 062 Energy-related parameters η SB 1 , 1 = 0 . 371 , η SB 1 , 2 = 0 . 212 , η SB 1 , 3 = 0 . 402 , η DB = 0 . 015 η SB 1 , 1 = η SB 1 , 2 = 0 . 142 , η SB 1 , 3 = 0 . 085 , η DB = 0 . 631 η SB 2 , 3 = 0 . 724 , η DB 2 , 1 = η DB 2 , 2 = 0 . 138 η SB 2 , 3 = 0 . 056 , η DB 2 , 1 = η DB 2 , 2 = 0 . 472 Envir onment-related parameters k (1 , 1) = 8 . 9 , k (1 , 2) = 2 . 7 , k (1 , 3) = 12 . 3 k (1 , 1) = 0 . 55 , k (1 , 2) = 1 . 21 , k (1 , 3) = 12 . 3 k (2 , 1) = 8 . 9 , k (2 , 2) = 2 . 7 , k (2 , 3) = 12 . 3 k (2 , 1) = 0 . 55 , k (2 , 2) = 1 . 21 , k (2 , 3) = 12 . 3 Although the MT and MR in the pr o posed model are employed in ULA o mni-direc tio nal antenna elements, the propo sed model can also be used to a n alyze ra diation p atterns speciﬁc to the elem ents, which m ake the prop osed geometric channel mo del irregularly shaped. Here , we assume that the transmitter emits the sign al to th e r eceiv er in signiﬁcan tly small beam widths, spa nning th e azimuth ran ge o f [ − α, α ] . It is stated in [ 16] th at the AoA statistics o f the multi-path compon ents are very useful in the p erform ance e valuation of wireless commu n ication systems employing MIM O a ntenna arrays at th e MT and MR. Here , the marginal PDF of the AoA statistics co rrespond ing to the road width b 1 and the beamwidths of th e directional an tenna ( i.e., α ) at the MT is shown in Fig. 5 . It is a pparent that, when the MT is employed with the direction al antenn a elem ents, th e AoA PDFs in 0 ≤ α ( n l, 3 ) R ≤ π ﬁrstly decrease to a local value of AoA and the n inc rease to a local maximu m with a “corner” , the AoA PDFs ﬁnally decrease sharp ly , dep ending upon the propo sed geometric chann el model, as seen in Figs. 1 an d 2. A similar behavior can be seen in − π ≤ α ( n l, 3 ) R ≤ 0 . By increasing the beamwidth s α with more scatterers in the scattering region illumin ated by th e directiona l antenna, the PDFs ﬁrstly have higher values on both sides of the curves, and then gradu ally tend to be equal. It can also be no ted that when the road wid th b 1 increases f rom 40 m to 80 m, the values of the AoA PDFs in c rease sharply . By adop ting an MT antenna element spacing δ T = λ , th e absolute values of the time - variant space CF of the pr o posed V2V chann el mod e l are illu strated in Figs. 6, 7, and 8. By imposing i = 1 and 3 in (23 ) , Fig. 6 shows the absolute values of time-variant space CFs of the single-bou nced models (i.e., SB 1 , 1 and SB 1 , 3 ) for different tran smit anten na azimuth angles ψ T and elevation ang le θ T . It is obvious th a t the spatial correlation gradu ally decr eases whe n th e normalized antenna spacing d · λ − 1 increases. A similar behavior can be seen in [28]. Additionally , it is evident that the value of the time- variant space CF g radually decreases as th e tra n smit an tenna angles (i.e., ψ T and θ T ) decr e ase. Figs. 7 an d 8 illustrate the absolu te values of th e time- variant space CFs for different cha n nel con ditions, i.e., WSS 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Normalized antenna spacing δ R / λ Normalized space CF SB 1,1 , ψ T = π /6, θ T = π /6 SB 1,1 , ψ T = π /3, θ T = π /3 SB 1,3 , ψ T = π /6, θ T = π /6 SB 1,3 , ψ T = π /3, θ T = π /3 Reference [28] Fig. 6. Absolut e v alues of time-v ariant space CFs of the single-bounce d models for dif ferent transmit antenna angles in tap one highway scenarios. and n on-WSS assump tions. By using (24 ) , the absolute values of the time- variant space CFs of the ﬁrst and secon d taps of the single-bou nced semi-ellipsoid model ( i.e., SB l , 3 ) f or different taps and d ifferent relative moving pro p erties (i.e., t a n d γ R ) are shown in Fig. 7. I n this ﬁgure, the higher correlation in the ﬁrst tap is com pared to the second tap b ecause of th e dom in ant Lo S rays, wh ic h is in correspo n dence with the results in [22]. By using (25) and im posing i = 1 and 3 in (23), Fig. 8 illustrates the absolute values of the time-variant spa c e CFs of the single- (i.e., SB 1 , 1 and SB 1 , 3 ) an d double - bounc e d models (i.e. , DB ) of the ﬁrst tap in the WSS co ndition (i.e. , t = 0 ). The ﬁgure shows th at the relativ e moving directio n s (i.e., γ R ) hav e no impact on th e d istribution of th e time-variant space CFs wh en the propo sed chan nel model is under th e WSS assumption. It can be observed that the time-variant space CF of the single- bounc e d SB 1 , 3 is lo wer than th a t o f the single-boun ced SB 1 , 1 . This is due to the fact that hig her ge o metric p a th len gths result in lower co rrelation as mentio ned in [27]. Howev er , in the propo sed m odel, th e p a th length for SB 1 , 3 is obviously lon ger than the path len gth for SB 1 , 1 . Fig. 9 shows the time-variant space CFs fo r the sing le- 11 0 0.5 1 1.5 2 2.5 3 3.5 4 Normalized antenna spacing δ R / λ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Normalized space CF Tap 1 , t = 0 (WSS) Tap 1 , t = 2s (non-WSS), γ R = π /3 Tap 1 , t = 2s (non-WSS), γ R = 2 π /3 Tap 2 , t = 0 (WSS) Tap 2 , t = 2s (non-WSS), γ R = π /3 Tap 2 , t = 2s (non-WSS), γ R = 2 π /3 Reference [22] Fig. 7. Absolute values of the time-va riant space CFs of the single-bounced semi-elli psoid model for diffe rent taps of the proposed model in highway scenari os. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Normalized antenna spacing δ R / λ Normalized space CF SB 1,1 , γ R = 0 SB 1,1 , γ R = π DB , γ R = 0 DB , γ R = π SB 1,3 , γ R = 0 SB 1,3 , γ R = π Fig. 8. Absolute val ues of the time-v ariant space CFs of the single- and double-b ounced m odels for dif ferent relati ve moving direct ions in highway scenari os. bounc e d rays of grou nd reﬂection with respect to the different antenna heigh ts (i.e., H t and H r ) an d different distances D between the MT and MR. Fro m the ﬁgure, we can easily notice that when the heights of antenna s moun ted o n the MT and MR incre ase fro m 10 m to 2 0 m , th e space CFs decre a se slowly , irrespective of the h ighway an d urb an environments. Additionally , the space CFs d e c rease g radually as the MR gets away fro m the MT . This is mainly d ue to the fact th at higher geometric path lengths result in th e lo wer correlation, as in Fig. 8. For V2V scena r ios, it is important to a nalyze the im pact of non-station arity , includ ing that of the relative moving di- rections (i.e., γ R ) and moving time instants (i.e., t ), on the statistical properties o f the proposed V2V chan nel model. According ly , by using (25) and (27) , Fig . 1 0 shows the time-varying fre quency CFs of the d ouble-b ounced models (i.e., DB a n d DB 2 , 2 ) co rrespond ing to the different relati ve moving direction s and different movin g time instants. It is clearly observed th at, for the doub le - bounc e d DB WSS mo del, 0 0.5 1 1.5 2 2.5 3 3.5 4 Normalized antenna spacing δ R / λ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Normalized space CF H t = H r = 10 m , D = 40 m H t = H r = 10 m , D = 60 m H t = H r = 20 m , D = 40 m H t = H r = 20 m , D = 60 m Fig. 9. Absolute values of the time-v ariant space CFs for the single-bounce d rays of ground reﬂection for dif ferent ant enna heights (i.e., H t and H r ) and dif ferent distances D between the MT and MR. 0 5 10 15 20 Frequency separation, ∆ f [MHz] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Normalized time-varying frequency CF t = 0 (WSS) , DB , γ R = π /3 t = 0 (WSS) , DB , γ R = 2 π /3 t = 2s (non-WSS) , DB , γ R = π /6 t = 2s (non-WSS) , DB , γ R = π /3 t = 2s (non-WSS) , DB 2,2 , γ R = π /6 t = 2s (non-WSS) , DB 2,2 , γ R = π /3 Fig. 10. Absolute value s of the time-v ariant frequenc y CFs of the double- bounced models for dif ferent relati ve moving directio ns and diffe rent time instant s in highway scenarios. regardless of what the relative moving directio ns are (i.e. , γ R = π / 3 or 2 π/ 3 ), the curves of the frequ ency CFs between them tend to be the same , which co nﬁrms the analy sis in Fig. 6. Furtherm ore, it is evident th at w h en the recei ver’ s relativ e movin g d ir ection γ R is π / 3 , the v alue o f the time- variant frequ ency CF is relatively h igher than that at γ R = π / 6 . This is bec a use higher geo metric path lengths result in lower correlation , whereas the path leng th for the p ath length at γ R = π / 3 is obvio usly shorter th an in th e oth er c ases [27]. Th en, we o bserve that the f requency CF of the d ouble- bounc e d DB 2 , 2 is lower than th at of the doub le-boun ced DB in the p roposed non -stationary V2V cha n nel m odel. These results well align with those of pr evious work [12] and thus demonstra te the utility of our mo d el. T o under stand the impact of the chan nel model parameter s and non -stationary prop erties on Doppler PSDs given in (3 5 ) for the theo retical m odel, Fig. 11 shows the no rmalized Doppler PSDs of the proposed V2V chann el model fo r dif- ferent relativ e moving direc tions. I t is observed th a t, f o r the 12 −100 −50 0 50 100 −50 −40 −30 −20 −10 0 Normalized Doppler frequency, Hz Normalized Doppler PSD (dB) SB 1,1 , γ R = π /3 SB 1,1 , γ R = 2 π /3 SB 1,2 , γ R = π /3 SB 1,2 , γ R = 2 π /3 SB 1,3 , γ R = π /3 SB 1,3 , γ R = 2 π /3 Fig. 11. Normalized Dopple r PSDs of the proposed V2V chan nel model for dif ferent relati ve mo ving directio ns in highwa y scenarios. -100 -80 -60 -40 -20 0 20 40 60 80 100 Normalized Doppler frequency [Hz] -50 -40 -30 -20 -10 0 Normalized Doppler PSD (dB) Tap 1 , SB 1,3 , γ R = 0 Tap 1 , SB 1,3 , γ R = π /2 Tap 1 , SB 1,3 , γ R = π Tap 2 , SB 2,3 , γ R = 0 Tap 2 , SB 2,3 , γ R = π Avazov Model [2] Clarke Model [14] Fig. 12. Normalized Doppler PSDs of the single-bou nced chann el model for dif ferent taps and differen t relati ve m oving direc tions in highway scenarios. direction of γ R = π / 3 , th e Doppler PSD o f the single-bou n ced SB 1 , 1 is larger th an that of the single-b ounced SB 1 , 3 because of the h igher fadin g loss cau sed b y the lon ger geo metric path length . It is also evident that, f o r the waves tha t are single-bou nced at the MR (i.e., SB 1 , 2 ), th e relative movin g direction has no impac t on the distribution of th e Dopp ler PSD. Moreover , th is Doppler distribution tends to be a c o n ventional U-shaped distribution, as shown in [10] . Moreover, com par- isons b etween th e ab ove th eoretical d iscussions with the Jiang model [5] show th a t the respective distribution trend s ar e in agreemen t, wh ich validates the generalization of the proposed V2V channe l m o del. Fig. 12 shows the normalize d Doppler PSDs of the single - bounc e d chan nel mod els (i.e., SB 1 , 3 and S B 2 , 3 ) f or different taps and d ifferent relative moving d ir ections (i.e., γ R ). It is observed that the Dop p ler freq uency gr adually decr eases with a decrease in the taps of the p roposed channe l mo del. It is also appar ent that, for the MR m ovement perp endicular to the direct LoS rays ( i.e., γ R = π / 2 ), each cur ve of the Dop p ler frequen cy of the con ventional stationary V2V chann el mode l tends to be in the A vazov model [2] with a p e a k at zer o . 1 1.2 1.4 1.6 1.8 2 Time Delay, τ ' (s) 0 0.2 0.4 0.6 0.8 1.0 Normalized Impulse Response Tap 1 , SB 1,1 , R t = 20m (Urban) Tap 1 , SB 1,1 , R t = 40m (Highway) Tap 1 , SB 1,3 Tap 2 , SB 2,3 Fig. 13. Absolut e value of the impulse response of the proposed single- bounced channel m odel for differe nt taps and diffe rent V2V scenarios. Howe ver, this is no t necessary fo r the pr oposed non- stationary V2V ch a nnel mod e l. W e thus conclud e that the curves o f the Doppler distribution constantly shift to the left region by increasing the re la tive moving time t when γ R is set π / 2 , as reported in [2 5 ]. In addition, if w e neglect the elev ation angles around the receiv er , the received signal comes from the single- bounc e d ra ys (i.e., SB 1 , 2 ) cau sed by the scatterers u niformly located on a circle ar ound the MR. Thus, the prop osed Do ppler PSD is g iven b y the classic Clarke sp ectrum, which alig ns with the results in [14]. Meanwhile, Fig. 13 illustrates the ab so lute value of the impulse response of the p r oposed 3D non - stationary V2V channel mod el fo r different time delay s. In the ﬁgu re, time delay τ ′ can be d eﬁned as the ratio of th e geometr ic path lengths and ligh t velocity c . The shortest and lo ngest prop- agation delay s of the propo sed WSS mo del are respectively obtained as τ ′ min = D / c and τ ′ max ≈ 2 a l / c . Furth e r more, it is evident that the impulse r esponse gradually decrea ses with an increase in time delay τ ′ , wh ich agr ees with the r e sults in [27]. In addition, the channel r esponse gradually d e creases with an increase in the taps o f the prop osed channel mod el, which is in agreemen t with the theore tica l analysis in Figs. 7 and 12. It is also apparent that the lower imp ulse channe l is R t = 2 0 m com pared to R t = 40 m becau se of the faster chan nel fading. The analysis above agre es with the results repor te d in [21], which can thus be fu lly utilized for the future design of wireless com munication systems. V . C O N C L U S I O N In this pap er , we hav e proposed a 3D wideba n d geometry- based chann el model for V2V comm unication scenarios. The relativ e movement b etween the MT and MR results in the time-variant geo metric statistics th at make our model no n - stationary . The p roposed model ad opts a two-cylind er mod el to d epict movin g vehicles (i.e., aroun d the MT or MR), as well as mu ltiple confoca l semi- ellipsoid models to mim ic stationary roadside en vironmen ts. Based on experim ental results, these channel statistics sho w different beh aviors at dif ferent r e lati ve moving time in stants, thereby demonstrating th e capability of 13 the prop osed mode l in depictin g a variety of V2V scenarios, i.e., macro -, micro -, and pico-cells. It is addition ally shown that the dominan ce o f the L oS compon ent results in a higher correlation in the ﬁrst tap of the pr oposed channel model than in the second o ne. From th e nu merical results, we conclud e that the time-variant space CF an d freq uency CF are signiﬁcantly affected by th e d ifferent taps of the propo sed time-variant chann el m odel, the relative moving tim e s, an d the directions between the MT an d MR. Finally , it is shown that the propo sed mo del clo sely agr ees with the measured data, which validates the utility of ou r model. V I . A C K N O W L E D G E M E N T S The autho rs would like to thank Prof essors Hikmet Sari and Guan Gui, Department of T elecomm unication and Info rmation Engineer ing, Na njing Un i versity of Posts and T elecomm u nica- tions, China, for h elping u s complete this stud y succ e ssfully . The autho rs would also thank the anonymous r evie wers for their c onstructive c o mments, wh ich g r eatly h elped imp rove this paper . R E F E R E N C E S [1] DOCOMO 5G White Paper , “5G radio access: requiremen ts, concept and technol ogies, ” NTT DOCOMO, INC. , Jul. 2014. [2] N. A vazo v and M. 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A 3D Non-Stationary Wideband Geometry-Based Channel Model for MIMO Vehicle-to-Vehicle Communication System

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