Deterministic and Stochastic Simulators for Non- …...delay line (TDL) structure, and the effect of Doppler spectral power density (DPSD) on the model was also investigated. However,

China Communications • July 201818

Keywords: vehicle-to-vehicle; wideband channels; simulation model; statistical proper-ties

I. INTRODUCTION

Nowadays, since the intelligent transportation system and vehicle self-organizing network have achieved rapid growth, the research on vehicle communication channel is increasing day by day [1], [2]. Because of the character-istics of vehicle traveling at high speed and limited moving area, the vehicle-to-vehicle (V2V) communication system has a significant difference from the traditional cellular system. The biggest difference between the V2V com-munication systems and the cellular network in the transmission environment is both the transmitter (Tx) and receiver (Rx) are moving. In addition, there are a lot of scatterers around the Tx and Rx. Therefore, in order to facilitate our understanding of the unique V2V channel characteristics and design of the vehicle com-munication systems, it is desirable to conduct in-depth and detailed research on different scenarios of V2V multiple-input multipleout-put (MIMO) channels, and develop accurate yet easy-touse channel models.

Based on our research [3], the V2V chan-

Abstract: In this paper, we consider a novel two-dimensional (2D) geometry-based sto-chastic model (GBSM) for multipleinput mul-tiple-output (MIMO) vehicle-to-vehicle (V2V) wideband fading channels. The proposed model employs the combination of a two-ring model and a multiple confocal ellipses model, where the signal is sum of the line-of-sight (LoS) component, single-bounced (SB) rays, and double-bounced (DB) rays. Based on the reference model, we derive some expressions of channel statistical properties, including space-time correlation function (STCF), Dop-pler spectral power density (DPSD), envelope level crossing rate (LCR) and average fade duration (AFD). In addition, correspond-ing deterministic and stochastic simulation models are developed based on the reference model. Moreover, we compare the statistical properties of the reference model and the two simulation models in different scenarios and investigate the impact of different vehicular traffic densities (VTDs) on the channel statis-tical properties of the proposed model. Finally, the great agreement between simulation mod-els and the reference model demonstrates not only the utility of simulation models, but also the correctness of theoretical derivations and simulations.

Deterministic and Stochastic Simulators for Non-Isotropic V2V-MIMO Wideband ChannelsYiran Li1 , Xiang Cheng1,*, Nan Zhang2

1 State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Electronics Engineering and Computer Science, Peking University, Beijing 100876, China2 Wireless Algorithm Department, Product Research and Development System, ZTE Corporation, Shanghai 201203, China* The corresponding author, email:[email protected]

Received: Mar. 21, 2018Revised: May 3, 2018Editor: Rongqing Zhang

5G ENABLED VEHICULAR COMMUNICATIONS AND NETWORKING

China Communications • July 2018 19

proposed a two-concentric-cylinder model for wideband channels, which can be considered as an extension of the twocylinder narrowband model. In [15], a 2D two-ring and ellipse mod-el for V2V narrowband channel has been ex-tended to wideband channel by using confocal multiple ellipses model to represent the tapped delay line (TDL) structure, and the effect of Doppler spectral power density (DPSD) on the model was also investigated. However, it did not derive the expression of the corresponding statistical properties of space-time correlation function (STCF), envelope level crossing rate (LCR) and average fade duration (AFD). Note that these aforementioned GBSMs [13]–[15] are the so-called reference model since these models assume an infinite number of effective scatterers, i.e., has an infinite complexity, and thus cannot be directly implemented in prac-tice. Unlike the reference model, a simulation model has a finite complexity and thus is real-isable in practice. Therefore, accurate simula-tion models play an important role in the prac-tical simulation and performance evaluation of any wireless communication systems. It is worth noting that the 2D GBSM modeling and investigation of V2V-MIMO wideband chan-nels are surprisingly missing in the current literature.

To fill up the aforementioned gaps of V2V-MIMO GBSMs, this paper proposes a new generic 2D GBSM for V2V-MIMO wide-band channels. The proposed 2D wideband GBSM mainly investigates on the basis of a two-ring and a multiple confocal ellipse model for V2V-MIMO wideband channel proposed in [15], which combines line-of-sight (LoS) components, singlebounced (SB) and dou-ble-bounced (DB) components. This paper’s main contributions and novelties can be sum-marized as follows.

1) Based on a proposed reference model for V2V-MIMO wideband channels, the ex-pressions of STCF, DPSD, envelope LCR and AFD are derived.

2) Corresponding deterministic and sto-chastic simulation models are developed based on the reference model. The great agreement

nel models can be divided into two categories generally, such as deterministic models [4] and stochastic models, and the latter can be further classified as non-geometrical stochastic mod-els (NGSMs) and geometry-based stochastic models (GBSMs). The GBSM uses simplified ray tracing principles and equivalent scatterer concepts to simulate propagation environ-ments. What’s more, the GBSMs can be clas-sified as two parts, including irregular shaped GBSMs (IS-GBSMs) [5] and regular-shaped GBSMs (RS-GBSMs). Since V2V commu-nication scenarios are in general time variant due to the movement of the Tx and Rx, the GBSM method is more suitable as this method directly deals with propagation environments. Therefore, the use of the GBSM method for modeling V2V propagation channels has at-tracted more and more attention.

In [6], in order to model the isotropic sin-gle-input singleoutput (SISO) V2V Rayleigh fading channel, a 2D two-ring RS-GBSM was proposed. In [7] and [8], the authors proposed a general two-ring GBSM, which is suitable for the non-isotropic V2V-MIMO Ricean channels. In addition, a RS-GBSM was proposed which is the combination of a ellipse and two-ring model for non-isotropic V2V-MIMO channels in [9] and [10]. The pa-per [11] proposed a generic three-dimensional (3D) twocylinder model for narrowband V2V channels. Furthermore, the paper [12] pro-posed some research on the vehicular traffic density (VTD), which can demonstrate the VTD have some effect on the channel statisti-cal characteristics of V2V channels.

Moreover, all the above mentioned RS-GB-SMs were proposed for the narrowband V2V channels. An important characteristic of the narrowband channel is that its propagation delay is far less than the data symbol duration Ts, the delay differences caused by different effective scatterers can be neglected. However, since the propagation delay is larger than the data transmission rate Ts and can not be ig-nored in the wideband systems. Therefore, it is important to develop the corresponding wide-band V2V channel models. Paper [13], [14]


ed around the ring of radius RR at Rx, and the n1,2th (n1,2 = 1, 2, ..., N1,2) effective scatterer is denoted by s( )n1,2 . In addition, for the elliptical model, the multiple confocal ellipses model is used here to represent the TDL structure, where their focal points are located at Tx and Rx. It can be seen that the distance between the transmitter and receiver can be expressed as D = 2f, which the parameter f represents the half length of the two foci connection of the ellipse. Assuming that the the lth ellipse’s semi-major axis (i.e., the lth tap) is al and there are Nl,3 effective scatterers distributed over it, (l = 1, 2, ..., L), where L is the total number of confocal ellipses. Note that the nl,3th (nl,3 = 1, 2, ..., Nl,3) effective scatterer can be described as s( )nl ,3 .

Fig.1 shows the basic 2×2 V2V-MIMO wideband channel model (nT = nR = 2), that is, the transmitter and receiver set up two uniform linear antenna components. In addition, the velocities of the Tx and Rx are denoted as vT and vR, and the movement direction angles are gT and gR, respectively. By considering the real V2V communication scenario, it is assumed that the moving scatterers distributed around Tx and Rx move at speed vST and vSR, respec-tively, and their movement directions are along the x-axis. The angle of departure (AoD) αT

( )nl i, ( 1, 2,3)i = characterizes the relative po-sition of scatterer S ( )nl i, to Tx. Similarly, the angle of arrival (AoA) αR

( )nl i, characterizes the relative position of scatterer S ( )nl i, to Rx. The AoD and AoA of the LoS path are denoted by αT

LoS and αRLoS .

Because the different rays have different contributions in a real V2V communication environments, we need to design different taps of our model to express the V2V channel statistics. For the first tap, the corresponding single- and double-bounced components can be considered as produced by the scatterers located on the first ellipse or one of the two rings, and the scatterers located on both two-rings, respectively. In addition, considering other taps, we design that only the scatterers on the confocal ellipses can generate the

between the reference model and simulation models demonstrates the correctness of deriva-tions and the utility of the proposed simulation models.

3) Based on the derived expressions of the V2V-MIMO wideband channel statistical properties, we investigate the impact of differ-ent VTDs on the channel statistical properties, and present some interesting observations and useful conclusions.

The structure of this paper can be summa-rized as follows. Section II introduces the ref-erence model of V2V-MIMO wideband chan-nel briefly. In Section III, from the reference model, the expressions of channel statistical characteristics are derived, including STCF, DPSD, envelope LCR and AFD. Based on the proposed reference model, corresponding deterministic and stochastic simulation models are proposed in Section IV. Section V presents some comparison and analysis of the statistical characteristics between proposed simulation models and the reference model. Finally, we draw some conclusions in Section VI.

II. A WIDEBAND V2V-MIMO CHANNEL REFERENCE MODEL

In this section, we propose a 2D GBSM for a wideband V2V-MIMO communication chan-nel based on [15]. It is assumed that both the Tx and the Rx that equipped with nT transmit and nR receive low elevation antenna elements are moving, where 1 ≤ p ≤ p’ ≤ nT and 1 ≤ q ≤ q’ ≤ nR, respectively.

It is assumed that the scatterers are distrib-uted over the tworing model and the confocal multi-ellipsoidal model randomly. The two-ring model is used to represent moving scat-terers, such as the moving vehicles, and the multiple confocal ellipse models are used to represent the static scatterers, such as the static roadside environment. For a two-ring model, it is assumed that N1,1 effective scatterers are dis-tributed around the ring of radius RT at the Tx, and the n1,1th (n1,1 = 1, 2, ..., N1,1) effective scatterer is defined as s( )n1,1 . Similarly, suppose there are N1,2 effective scatterers are distribut-


maximum Doppler frequency because the transmitter, receiv-er and some scatterers are moving. In addition, ηSBi1

and ηDB

designate the contribution of the single- and double-bounced components to the total scattered power Wpq/(K + 1), which

satisfy ∑i1

I

=

1

1η ηSB DBi1

+ =1 . For a low VTD, because most of

the total power comes from the LoS component, the param-eter K is large. What’s more, since the number of moving scatterers is small, the static scatterers located on the first ellipse are allocated a large amount of power, and the propor-tion of the double-bounced components is smaller than that of the single reflected components. This can be express as η η η ηSB SB SB DB1,3 1,1 1,2

> >max , . In other side, the parameter K

is small in a high VTD scene. What’s more, since the amount of moving cars is large, the singlebounced components from the ellipse and two-ring models are allocated less energy than the double-bounced components from the two-ring models, which can be express as η η η ηDB SB SB SB> max , ,

1,1 1,2 3 .

For other taps (l’ > 1), the complex tap coeffi cient h tl pq', ( ) can be expressed as

h t h t h tl pq l pq l pq', ', ',( ) ( ) ( )= +SB ∑i2

I

=

2

1

DBi2 (3)

with

h tl pq SB pqSB', ( ) lim= Ω

e

∑N

nl

l

',3

j f t f t

',3

[2 cos( ) 2 cos( )]

η

e e

π α γ π α γ

jφ

Tm T Rm R

( )

l

n

',3

l ',3 −

T R( )

2

nl l

π τ

N

',3 ',3

l

f

',3

c pq n

− + −

→∞

, l ',3

N1

l ',3

( )n

(4a)

single-bounced rays, while the the dou-ble-bounced rays come from the correspond-ing ellipse and one of the two rings.

H e r e , w e u s e a n n T × n R m a t r i x H t h t( , ) [ ( , )]τ τ= pq n nR T× t o d e s c r i b e t h e V2V-MIMO wideband channel. The channel impulse response between the pth Tx and the qth Rx antenna elements can be expressed as h t c h tpq l l pq l( , ) ( ) ( )τ δ τ τ= −∑ l

L

=1 , , where cl rep-

resents the gain of the lth tap, and the complex time-variant tap coefficient and the discrete propagation delay of the lth tap are denoted as h tl pq, ( ) and τ l , respectively. For the fi rst tap, the complex tap coeffi cient of the Tp-Rq link can be expressed as [16]

h t h t h t h t1, 1, 1, 1,pq pq pq pq( ) ( ) ( ) ( )= + +LoS DB∑i1

I

=1

SBi1 (1)

with

h t e1,LoS

pq ( ) =

e j f f[2 cos( ) 2 cos( )]π τ α γ π τ α γ

KK

Tm T T Rm R R

Ω

+pq

1−

LoS LoS

2π τ

− + −

fc pq (2a)

h t1,SB

pqi1 ( ) lim=

e

e ej f t f t

jφ

[2 cos( ) 2 cos( )]

( )

η

n

π α γ π α γ

1,i1

SB pq

K

Tm i T Rm i R

1,

, ,

i

1 1

1

−

+

Ω

2π τ

1fc pq n

T R( ) ( )n n

N

1, 1,

,

1,

i i1 1

1,

i1

l1

− + −

→∞∑N

n1,

1,

l1

l1

N1

1,l1

(2b)

h t1,DB

pq ( ) lim=

e

N N

n n1,1 1,2

1,1 1,2

∑j f t f t[2 cos( ) 2 cos( )]

,

,

η

π α γ π α γ

KDB pq

Tm T Rm R

e e

,1 ,2

Ω

+

jφ

1

( , )n n1,1 1,2

T R

N N

( ) ( )n n

1,1 1,2

1,1 1,2

,

− + −

−2π τ

→∞

fc pq n n, ,1,1 1,2

N N1,1 1,2

1

(2c)

where I1 = 3, the symbols Wpq and K de-note the total power of the Tp-Rq link and the Ricean factor, respectively. The scatter-ing-caused phases φ ( )n1,i1 and φ ( , )n n1,1 1,2 are ran-do variables with uniform distributions over [-π, π). τ εpq pq= / c , τ ε εpq n pn n q, 1, 1, 1,l l l1 1 1

= +( ) / c ,

and τ ε ε εpq n n pn n n n q, 1,1, 1.2 1,1 1,1 1,2 1,2= + +( ) / c are the

waves’ travel times through the link Tp Rq− , Tp S Rq− −( )n1,i1 , and Tp S S Rq− − −( ) ( )n n1,1 1,2 , respectively, and c is the speed of light. fT(R)m = vT(R)/λ, fT(R)m,1 = |vT(R) cosg T(R)-vST|/λ, fT(R)m,2 = |vT(R) cosg T(R)-vSR|/λ, and fT(R)m,3 = vT(R)/λ are the

Fig. 1. A RS-GBSM combining a two-ring model and a multiple confocal ellips-es model with LoS components, single- and double-bounced rays for wideband V2V-MIMO channels (nT = nR = 2).

1,1( )ns1,2( )ns

1,3( )ns

2,3( )ns

RRTR

2D f

12a22a

p

'p

q

'q

1,1( )nT

1,3( )nT

1,2( )nT

2,3( )nT

1,3( )nR

2,3( )nR

1,2( )nR

1,1( )nRT

TvRv

RT

R

STvSRv


over the theoretical reference model are usu-ally assumed to be infinity, we can use the the continuous expressions of AoD αT

l i, and AoA αR

l i, to replace the discrete expressions αT( )nl i.

and αR( )nl i, , respectively. Based on our research,

the distribution of azimuth angles for scatter-ers αT

( )nl i. and αR( )nl i, has been described using

several probability density functions, includ-ing uniform [17], Gaussian, Laplacian [18], and von Mises [19]. But the most commonly used is the von Mises PDF because it approx-imates many of the previously mentioned dis-tributions and leads to closed-form solutions for many useful situations [11]. It is defined as f k I k( ) exp[ cos( )] / 2 ( )α α α π= − u 0 , where α π π∈ −[ , ) , I0 ( ) is the zeroth-order modi-

fied Bessel function of the first kind, αµ ac-

counts for the mean value of the angle α , and the real-valued parameter k(k > 0) is designed to control the distribution of the angle αµ .

III. STATISTICAL PROPERTIES OF THE PROPOSED WIDEBAND V2V-MIMO CHANNEL MODEL

In this section, based on the research of litera-ture on the statistical properties of narrowband V2V-MIMO reference models [7] – [11], some important channel statistical characteristics of the V2V-MIMO wideband channel model for non-isotropic scattering environment will be derived, including the STCF, Doppler PSD, envelope LCR and AFD.

3.1 Space-time correction function

The normalized space-time CF [20] be-tween two arbitrary complex tap coeffi-cients h tpq ( ) and h tp q' ' ( ) can be defined as

Rpq p q T R, ' ' ( , , )δ δ τ =E h t h t[ ( ) ( )]*

pq p q

Ω Ωpq p q

' '

' '

+τ, where

E[ ] and ( )⋅ * denote the statistical expectation operator and complex conjugate operation, respectively. It can be written as the superpo-sition of the LoS, single- and double-bounced components. For the first tap,

h tl pq DB pqDB',

1 ( ) lim= Ω

e

N N

n n

1,1 ',3

1,1 ',3

∑j t f f2 [ cos( ) cos( )]

,

,

η

π α γ α γ

l

l

Tm T Rm R

e e

l ',1

,1

jφ ( , )n n1,1 ',3

T R( ) ( )

l

n n

N N

1,1 ',3

1,1 ',3

− + −

,

−2

l

π τ

→∞

fc pq n n, ,1,1 ',3

N N

l

1,1 ',3

1

l

l

(4b)

h tl pq DB pqDB',

2 ( ) lim= Ω

e

N N

n nl

l

j t f f

∑',3 1,2

',3 1,2

2 [ cos( ) cos( )]

,

,

η

π α γ α γTm T Rm R

e e

l ',2

jφ ( , )n nl ' 3 1,2

T R( ) ( )n nl ',3 1,2

N Nl ',3 1,2

− + −

,

−2π τ

→∞

fc pq n n

,2

, ,l ',3 1,2

N Nl

1

',3 1,2

(4c)

w h e r e I 2 = 2 , τ ε εpq n pn n q, l l l',3 ',3 ',3= +( ) / c ,

τ ε ε εpq n n pn n n n q, 1,1, ',3 1,1 1,1 ',3 ',3l l l= + +( ) / c , a n d

τ ε ε εpq n n pn n n n q, l l l',3 1.2 ',3 ',3 1,2 1,2= + +( ) / c a r e

the waves’ travel times through the link Tp S Rq− −( )nl ',3 , Tp S S Rq− − −( ) ( )n n1,1 ',3l , and Tp S S Rq− − −( ) ( )n nl ',3 1,2 , respectively. The en-ergy related parameters ηDBl i', 2

and ηSB satisfy

∑i2

I

=

2

1η ηDB SBl i', 2

+ =1 . For a low VTD, the static

scatterers located on the ellipse are allocat-ed a large amount of power, which indicates that η η ηSB DB DBl l l',3 ',1 ',2

> max , . In addition,

since the amount of moving vehicles is large under a high VTD, the single-bounced com-ponents from ellipse models are allocated less energy than the double-bounced components produced from the combination of one-ring and ellipse models, which can be express as min , η η ηDB DB SBl l l',1 ',2 ',3

> . The AoDs and AoAs

are discrete random variables, and they can be converted by the following widely used approx-imate relationships [15]. For two rings model,

α π αR T( ) ( )n n1,1 1,1= −

RD

T sin , α αT R( ) ( )n n1,2 1,2=

RD

R sin .

For the multiple confocal ellipses mod-

e l , sinαT( )nl ,3 =

a f a fl l R2 2+ +

bl R2 sin

2 cosα ( )nl ,3

α ( )nl ,3,

cosαT( )nl ,3 =

2 ( )cosa f a fa f a f

l l R2 2l l R

+ ++ +2 2

2 cosαα

( )n

( )

l

n

,3

l ,3

. And in

the case of the LoS components, αTLoS = 0 and

α πRLoS = . Due to page limitation, the specific

expressions of distance can follow the paper [9].

As we know, since the scatterers distributed


3.2 Doppler power spectral density

In order to study the corresponding Doppler PSD of the proposed V2V wideband model, we can use the Fouri-er Transf Rorm to the STCF, which can be expressed as

S f R e dpq p q D pq p q, ' ' , ' '( ) ( )== ∫−∞∞

τ τ− j f2π τD , where fD is the Dop-

pler frequency.

3.3 Envelope LCR

Here, based on the research [21], we can define the LCR L(r) using the rate at which the signal envelope crosses a specified level r with a positive or negative direction. The LCR for V2V channels can be written as

L r e( ) = − ×

[ sin ( sin )]∫e erf d0

π

2 1

−

/2

( sin )

r K

χ θ

cosh(2 ( 1) cos )π 3/2 2

2

+

+ ⋅

K K rπ χ θ χ θ θ

b bb b0 0

2 1

+ ⋅

2− − +K K r( 1)

θ

2

(9)

where cosh( )⋅ and erf ( )⋅ can be described as the hy-perbol ic cos ine funct ion and er ror funct ion , and

χ = −Kb b b b1 0 2 12 2/ ( ) . Based on the [10], we first discuss

the parameters b0 of the first tap of the proposed model, which can be expressed as

b b b b b0 0 0 0 0= + + + =SB SB SB1,1 1,2 1,3 DB

2( 1)K1+

. (10)

Similarly, we can express parameters b1 and b2 as b b b b b mm m m m m= + + + =SB SB SB1,1 1,2 1,3 DB , ( 1, 2) (11)

with

b fm TSB1, 1i =

[ cos( )

+ −

f2( 1)

T i T T

f dRm i R R T

η

m,

KSB

1

, 1

1,

+

cos( )]

i

α γ

(2 ) ( )

α γ α

1,

π α

1, 1,

i1

i i1 1

−

m ∫−π

π

m

1,i1

(12a)

b f fm T RDB m=

[ cos( )

+ −

f2( 1)

T T T

f d dRm R R T R

m,1

ηK

,2

DB

+

cos( )]

α γ

(2 ) ( ) ( )

α γ α α

1,1

π α α

1,2 1,1 1,2

−

∫ ∫− −

π π

π π

m

1,1 1,2

. (12b)

For other taps, we can obtain parameter b0 as

b b b b0 0 0 0= + + =SB DB DBl l l',3 ',1 ',2

2( 1)K1+

(13)

The parameters b1 and b2 can be expressed as b b b b mm m m m= + + =SB DB DBl l l',3 ',1 ',2 , ( 1, 2) (14)

with

b fm TSBl ',3 =

[ cos( )+ −

f2( 1)

T T T

f dRm R R T

η

m

KSB

cos( )]

l

+',3

α γ

α γ α

(2 ) ( )

l

l m l

',3

',3 ',3

π α

−

m l∫−π

π

',3

(15a)

+ +

R R1, , ' ' 1, , ' '

∑i1

I

=

pq p q T R pq p q T R

1

1R R1, , ' ' 1, , ' '

SBpq p q T R pq p q T R

( , , ) ( , , )

1,

δ δ τ δ δ τ

i1 ( , , ) ( , , )δ δ τ δ δ τ

= LoS

DB (5)

with

R e1, , ' '

LoSpq p q T R

e j f f2 [ cos( ) cos( )]πτ α γ α γ

( , , )δ δ τ

Tm T T Rm R RLoS LoS− + −

=K

K+1

− −j 2λπ ( )ε εpq p q' '

(6a)

R f1, , ' ' ( )SB

pq p q T R T R1,i1 1( , , ) ( )δ δ τ α=

e d

e

e− + − +

η

j f

j f

K

2 cos( )

2 cos( )]

j

πτ τ α γ

πτ τ α γ

2

SB

λπ

+1,

[( ) ( )]

Tm i T

Rm i R

i1

1ε ε ε ε

,

,

pn n q p n n q

1

1

∫1, 1, 1, 1,

−

π

i i i i1 1 1 1

π

T R1,

R T( )n

( )i

( )

1

1,i1

−

1,

−

i

' '

αT R1,i

( )1

(6b)

R f f1, , ' 'DB

pq p q T R T R( , , ) ( ) ( )δ δ τ α α=

e d d

e

ei f

− + − +

j f

K

2 cos( )

η

2 cos( )

j

πτ τ α γ

πτ α γ

2λ

DB

π

+

Rm R

[( ) ( )]

Tm T

1ε ε ε ε

,2

,1

pn n q p n n q

∫ ∫1,1 1,2 1,1 1,2

− −

π π

π π

T1,1

1,2R

−

−

' '

α α

1,1 1,2

T R1,1 1,2

.

(6c)For other taps,R Rl pq p q T R l pq p q T R', , ' ' ', , ' '( , , ) ( , , )δ δ τ δ δ τ=

+∑i2

I

=

2

SB

1Rl pq p q T R

DB', , ' '

l i', 2 ( , , )δ δ τ (7)

with

R fl pq p q T R SB RSB l', , ' ' ( , , ) ( )δ δ τ η α=

e d

e

e− + − +

j f

j f2 cos( )

2 cos( )

j

πτ τ α γ

πτ α γ

2λπ

l ',3

[( ) ( )]

Tm T

Rm R

ε ε ε ε

∫pn n q p n n q

−

π

l l

π

',3 ',3

Tl ',3

Rl ',3

l l

−

',3 ',3

−

',3

α

' '

Rl ',3

(8a)

R f fl pq p q T R DB T RDB', , ' '

l '1 ( , , ) ( ) ( )δ δ τ η α α=

e d d

e

e− + − +

j f

j f

2 cos( )

2 cos( )

j

πτ α γ

πτ α γ

2λπ

l

[( ) ( )]

',1

Tm T

Rm R

ε ε ε ε

∫ ∫

,1

pn n q p n n q

− −

π π

1,1 ',3 1,1 ',3

π π

Rl

T1,1

',3

l l

−

−

α α

1,1 ',3

' '

T R1,1 ',3l

l

(8b)

R f fl pq p q T R DB T RDB', , ' '

l ' 2 ( , , ) ( ) ( )δ δ τ η α α=

e d d

e

e− + − +

j f

j f

2 cos( )

2 cos( )

j

πτ α γ

πτ α γ

2λπ

l

[( ) ( )]

',2

Tm T

Rm R

ε ε ε ε

∫ ∫

,2

pn n q p n n q

− −

π π

l l',3 1,2 ',3 1,2

π π

Tl ',3

1,2R

−

−

' '

l

α α

',3 1,2

T Rl ',3 1,2

.

(8c)


4.1 Deterministic simulation model

First, a deterministic simulation model is proposed, which requires constant parame-ters during simulation. In other words, based

on our proposed model, the AoDs α~

T

( )nl i.

and

AoAs α~ ( )

R

nl i,

have definite values for different simulation experiments. Based on [24], the AoDs and AoAs are designed as the follows.

1) We first define a new parameter α~'T R( )n

( )l i. ,

which has the the same environment parame-

ters as α~

T R

( )n

( )

l i.

on a von Mises distribution.2) Then, we use the following method to

get the set of α~ 'T R( )n

( )l i,

n

N

l i

l i

,

,

=1

as

α~' ( ), ( 1, 2,..., )T R T R l i l i( )n

( ) ( ) , ,l i, = =F n N−1 nl i,

N−

l i

1/ 4

,

(17)

where FT R−( )1 denotes the inverse function of

the von Mises CDF for α~'T( )nl i. .

3) In order to ensure that the simulation model’s AoDs and AoAs in the range of [ , )−π π , we use the following mapping, that is

α α π α π~ ~ ~

T R

( )n

( )

l i,

= − >

α π α π

~ ~' 2 , 'T R T R( ) ( )

' 2 , '

n n

T R T R( ) ( )

( ) ( )l i l i

n n

, ,

( ) ( )l i l i, ,

+ < −

α~' ,T R( )n

( )l i, else

(18)

4.2 Stochastic simulation model

The deterministic simulation model is easy to implement and has a short simulation time. However, in an actual communication chan-nel, the scatterers are not placed in a certain place like the proposed deterministic model. Therefore, this model can be transformed into a stochastic simulation model if we allow the phases or frequencies to be random variables. Since the addition of random variables, the channel characteristics of the stochastic sim-ulation model change with each simulation trial, but will converge to the expectations of the model in a sufficient number of simulation trials. Similarly, the AoAs and AoDs designed by the stochastic simulation model can be ex-

b f fm T RDBl ',1 =

[ cos( )

+ −

f2( 1)

T T T

f d dRm R R T R

η

m,1

KDB

cos( )]

+l ',1

α γ α α

α γ

(2 ) ( ) ( )

l m l

1,1

',3 1,1 ',3

π α α

−

m l∫ ∫− −

π π

π π

1,1 ',3

(15b)

b f fm T RDBl ',2 =

[ cos( )+ −

f2( 1)

T T T

f d d

η

Rm R R T R

m

KDB

,2

+l

cos( )]

',2

α γ

(2 ) ( ) ( )

α γ α α

l ',3

π α α

1,2 ',3 1,2

−

m l∫ ∫− −

π π

π π

m l

',3 1,2

. (15c)

We can bring the b0, b1, and b2 into the (9) to get the LCR of the proposed model.

3.4 Envelope AFD

The average time when the signal envelope |hpq(t)| stays be-low a certain level r is used to represent the signal envelope AFD T (r) . In the proposed model, the AFD can be written as [10]

T r( ) =1 2 , 2 1− +Q K K r(

L r( )( ) 2 )

(16)

where Q( )⋅ is the Marcum Q function.

IV. NEW 2D WIDEBAND V2V-MIMO CHANNEL SIMULATION MODELS

The reference model considers an infinite number of scatter-ers, but an infinite number of sinusoidal curves can not be realized in reality. Thus, we need to design a corresponding simulation model with limited complexity and capable of being implemented in practice. Therefore, a limited number of scatterers are consider in the simulation model, and this model is intended for system performance evaluation for reasonable complexity [22], [23].

Based on the above discussion, the approximation of the channel statistical characteristics between the simulation model and the reference model depends on the sampling mode of the scatterers. When the scatterer’s sample ap-proaches the probability density function of the scatterer distribution in the reference model, the practicality of the simulation model will become stronger. In other words, it is important to find a way to design the sets of AoDs αT

( )nl i. and AoAs αR

( )nl i, of the simulation model to represent the desired channel statistical characteristics. Next, we will propose the corresponding deterministic and stochastic simulation mod-els, respectively.


three-tap model, which the semi-major axis of the confocal multi-ellipse is assumed to be aL = 160,170,180 . Suppose the envi-ronment-parameters are K = 3.8 , kT = 6.6 , µT =12.8 , kR = 8.3 , µR =178.7 , kE = 7.7 , µE = 31.3 for low VTD, and K = 0.856 , kT = 0.6 , µT =12.8 , kR =1.3 , µR =178.7 , kE = 8.5 , µE = 20.6 for high VTD. In the case of low VTD, for the fi rst tap, it is assumed that the corresponding energy-parameters are ηSB1,1

= 0.203 , ηSB1,2= 0.335 , ηSB1,3

= 0.411 ,

and ηDB = 0.051 . For the l’th taps (l’>1), assume that the corresponding energy pa-rameters are ηSBl ',3

= 0.762 , ηDBl ',1= 0.119 ,

and ηDBl ',2= 0.119 . In the case of high VTD,

for the first tap, we use energy-parameters ηSB1,1

= 0.126 , ηSB1,2= 0.126 , ηSB1,3

= 0.063 ,

and ηDB = 0.685 to describe the fi rst tap. For the l’th taps (l’>1), assume that the corre-sponding energy parameters are ηSBl ',3

= 0.088 ,

ηDBl ',1= 0.456 , and ηDBl ',2

= 0.456 . Both de-

terministic and stochastic simulation models assume that the number of effective scatterers is N1,1 = N1,2 = N1,3 = N2,3 = N3,3 = 20 and the

pressed as follows.1) We fi rst propose a new random variable

α∧

'T R( )n

( )l i. , which has the the same environment

parameters as α∧

T R

( )n

( )

l i.

on a von Mises distribu-tion.

2) Then, we use the following method to

get the set of α∧ 'T R( )n

( )l i,

n

N

l i

l i

,

,

=1

as

α∧

' ( ), ( 1, 2,..., )T R T R l i l i( )n

( ) ( ) , ,l i, = =F n N−1 nl i, + −

Nθ

l i,

1 / 4.

(19)The parameters θ is independent random

variable uniformly distributed on the interval [ 1/ 2,1/ 2)− . Due to the introduction of ran-dom variable , the set of AoDs and AoAs var-ies with different simulation trial.

3) In order to ensure that the simulation model’s AoDs and AoAs in the range of [ , )−π π , we use the following mapping, that is

α α π α π∧ ∧ ∧

T R

( )n

( )

l i,

= − >

α π α π∧ ∧

' 2 , 'T R T R( ) ( )

' 2 , '

n n

T R T R( ) ( )

( ) ( )l i l i

n n

, ,

( ) ( )l i l i, ,

+ < −

α∧

' ,T R( )n

( )l i, else

. (20)

Based on our proposed way, we can get dis-crete expressions of the parameters the AoAs and AoDs of 2D V2VMIMO simulation model and bring them into (1)–(4) to get the corre-sponding statistical properties. Because of page constraints, the corresponding statistical properties’ derivation of the simulation models can refer to Section III.

V. NUMERICAL SIMULATION AND ANALYSIS

In this section, we will compare the chan-nel statistical characteristics between the proposed simulation models and reference model, and validate the usefulness of the two simulation models. The values of the pa-rameters used for our numerical analysis are D=2f=300m, RT=RR=10m, γT=γR=0, fc=5.9GHz, fT(r)m,1=fT(r)m,2=360Hz, fTm=fRm=fT(r)m,3=570Hz. Due to page constraints, we only consider a

Fig. 2. Doppler PSDs of the reference model and the two simulation models with different VTDs for the same direction of movement of the Tx and Rx under two dif-ferent tap scenarios: (a) fi rst tap and (b) second tap.

0 200 400 600 800 1000 1200 1400 1600 1800 2000Frequency, f(Hz)

(a)

-5

0

5

10

15

Dop

pler

pow

er s

pect

rum

den

sity

0 200 400 600 800 1000 1200 1400 1600 1800 2000Frequency, f(Hz)

(b)

-5

0

5

10

15

Dop

pler

pow

er s

pect

rum

den

sity

referencedeterministicstochastic

low VTD

high VTD

high VTD

low VTD


directions, the Doppler PSDs between the pro-posed reference and simulation models with different VTDs, where the images (a) and (b) denote the fi rst and second tap. It can be seen that the proposed simulation model can match well with the reference model, which further proves the usefulness and practicality of our proposed deterministic and stochastic simu-lation models. At the same time, the VTDs also have some effect on the DPSD for V2V channels. It can be observed that the Doppler PSD distribution is more flatly under high VTD scenarios, while the DPSD tends to be steeply with low VTDs. It is because that in a low VTD scenario, the received power mainly concentrates on several directions, e.g., the static scatterers on the roadside environment and/or the directions of LoS components, which results in the received power tending to concentrate at certain Doppler frequencies. In contrast, in the case of high VTDs, the re-ceived power mainly comes from the moving cars around the Tx and Rx from all directions, which means the distribution of DPSD is more uniform.

Similarly, Figs. 4 and 5 show the LCR between the reference and simulation mod-els with different VTDs when the moving direction of Tx and Rx is same or opposite, where the images (a) and (b) present the fi rst and second tap, respectively. We can see that the value of LCR is small when the VTD is low. This phenomenon can be explained that in a high VTD, the received power mainly comes from moving scatterers distributed in all directions, and the rate of change of the signal envelope is high, resulting in the value of the corresponding LCR is large. Unlike the high VTD scenarios, the received power comes from several specific directions with low VTDs. Therefore, the temporal stability of V2V channels is higher, and the value of LCR shown in the image is low.

Fig. 6 shows the STCF of the proposed models with different VTDs when the moving direction of Tx and Rx is same and opposite. It can be seen that the VTD affects the STCF of the model. The space-time correlation de-

number of simulation experiments for stochas-tic simulation model is Nstat = 50.

Figs. 2 and 3 compare that when the Tx and Rx are moving in the same and opposite

Fig. 3. Doppler PSDs of the reference model and the two simulation models with different VTDs for the opposite direction of movement of the Tx and Rx under two different tap scenarios: (a) fi rst tap and (b) second tap.

Fig. 4. Envelope LCR of the reference model and the two simulation models with different VTDs for the same direction of movement of the Tx and Rx under two dif-ferent tap scenarios: (a) fi rst tap and (b) second tap.

0 200 400 600 800 1000 1200 1400 1600 1800 2000Frequency, f(Hz)

(a)

0

5

10

15

Dop

pler

pow

er s

pect

rum

den

sity

0 200 400 600 800 1000 1200 1400 1600 1800 2000Frequency, f(Hz)

(b)

0

5

10

15

Dop

pler

pow

er s

pect

rum

den

sity


low VTD

high VTD

low VTD

high VTD

-15 -10 -5 0 5R(dB)

(a)

0

0.2

0.4

0.6

0.8

Nor

mal

ized

leve

l cro

ssin

g ra

te

-15 -10 -5 0 5R(dB)

(b)

0

0.5

1

1.5

2

Nor

mal

ized

leve

l cro

ssin

g ra

te


low VTD

high VTD

low VTD

high VTD


2018ZX03001031.

creases as the VTD increases. This phenome-non can be interpreted as the spatial diversity of the channel increases as the VTD increases, and the correlation decreases. At the same time, it shows that both deterministic and stochastic simulation models can have a rea-sonable agreement with the reference model. In addition, we can observe that the stochastic simulation model can obtain better agreement with the reference model compared with the deterministic simulation model. Therefore, although the deterministic simulation model is easy to be implemented and can be achieved with a short simulation time, it is more suit-able to choose the stochastic simulation model for the system performance evaluation when the computational complexity is similar.

VI. CONCLUSION

In this paper, based on the study of a reference model for non-isotropic V2V-MIMO wide-band channels which is the combination of a two-ring and a multiple confocal ellipses mod-el, we have derived the expressions of channel statistical characteristics, i.e., STCF, DPSD, LCR and AFD. Moreover, corresponding de-terministic and stochastic simulation models have been proposed based on the reference model. Based on numerical results, we have found out that different VTDs have a signifi -cant impact on channel statistical properties, and the great agreement between the reference model and simulation models has demon-strated the correctness of derivations and the utility of proposed simulation models. Finally, the simulation results have shown that the sto-chastic simulation model is more suitable for the system performance evaluation when the computational complexity is similar.

ACKNOWLEDGEMENTS

This work was supported in part by the project from the ZTE, the National Natural Science Foundation of China under Grant 61622101 and Grant 61571020, and National Science and Technology Major Project under Grant

Fig. 5. Envelope LCR of the reference model and the two simulation models with different VTDs for the opposite direction of movement of the Tx and Rx under two different tap scenarios: (a) fi rst tap and (b) second tap.

Fig. 6. Space-Time correlation function of the reference model and the two simu-lation models with low and high VTDs when the Tx and Rx move (a) in the same direction and (b) in the opposite direction.

-15 -10 -5 0 5R(dB)

(a)

0

0.5

1

1.5

Nor

mal

ized

leve

l cro

ssin

g ra

te-15 -10 -5 0 5

R(dB)(b)

0

0.5

1

1.5

Nor

mal

ized

leve

l cro

ssin

g ra

te


low VTD

low VTD

high VTD

high VTD

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Time Separation, t(s)

(a)

0.5

1

1.5

2

2.5

3

Spac

e-Ti

me

CF

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Time Separation, t(s)

(b)

0.5

1

1.5

2

2.5

3

Spac

e-Ti

me

CF


low VTD

low VTD

high VTD

high VTD


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Nan Zhang, received the bachelor degree in communi-cation engineering and the Master degree in integrated circuit engineering from Tongji University, Shanghai, China, in July 2012 and March 2015, re-spectively. He is now a Senior

Engineer at the Department of Algorithms, ZTE Cor-poration. His current research interests are in the field of 5G channel modeling, new air-interface and MIMO techniques. Email: [email protected]

BiographiesYiran Li, graduate student with the School of Software and Mi-croelectronics, Peking Universi-ty. Her current research inter-ests include channel modeling and vehicular communications. Email: [email protected]

Xiang Cheng, professor with the School of Electronics Engi-neering and Computing Sci-ence, Peking University. His current research interests in-clude channel modeling, wire-less communications (vehicular communications and 5G), and

data analytics. Email:[email protected]

Deterministic and Stochastic Simulators for Non- …...delay line (TDL) structure, and the effect of Doppler spectral power density (DPSD) on the model was also investigated. However,

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