a non-stationary mimo channel model ... - southeast university

2017 IEEE/CIC International Conference on Communications in China (ICCG)

A Non-Stationary MIMO Channel Model for StreetCorner Scenarios Considering Velocity Variations of

the Mobile Station and Scatterers

Ji Bian l , Yu Liu l , Cheng-Xiang Wang2 , Jian Sun", Wensheng Zhangl , and Minggao ZhangI

I Shandong Provincial Key Lab of Wireless Communication Technologies,School of Information Science and Engineering, Shandong University, Jinan, 250100, China.

2Institute of Sensors, Signals and Systems, School of Engineering & Physical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK.Email: {bianjimail.xinwenliuyu}@[email protected].{sunjian.zhangwsh}@[email protected]

Abstract-Most channel models in the literature assume thatthe scatterers are fixed and the mobile station (MS) moveswith a constant speed in a given direction. However, in realisticpropagation environments, both the scatterers and the MS canbe moving, and the velocities of the scatterers and the MS canchange with time. In this paper, we develop a non-stationarymultiple-input multiple-output (MIMO) channel model for streetcomer scenarios. The proposed channel model takes into accountboth fixed and moving scatterers. Velocity variations, includingspeed and movement direction, of the MS and moving scatterersare considered. Analytical solutions of spatial cross-correlationfunction (CCF), temporal autocorrelation function (ACF), andWigner-Ville spectrum are derived and analyzed. Moreover, theimpacts of velocity variations on the statistical properties of theproposed model are investigated. The proposed channel modelis illuminating for future vehicle-to-infrastructure (V2I) andvehicle-to-vehicle (V2V) channel modeling.

Index Terms-Non-stationary MIMO channel model, velocityvariations, time-variant parameters, statistical properties, streetcomer scenarios.

I. INTRODUCTION

The V2V and V2I communications have drawn increasingattention from researchers in recent years and are consid-ered to playa crucial role in the next-generation intelligenttransportation systems (ITSs) [1], [2]. The V2V and V2Icommunications are assumed to provide many benefits, suchas reducing traffic accidents and improving traffic efficiency.For example, in street comer scenarios, as illustrated in Fig. 1,drivers usually have poor visibility of the road. Real-time roadcondition information provided through extremely low latencyV2IN2V communications can help to avoid a collision. Forsystem design and performance evaluation, detailed and accu-rate knowledge of the channels for street corner scenarios isnecessary.

Most existing channel models are assumed to satisfy thewide-sense stationary (WSS) assumption, which means thestationarity of the channel during the observation time intervalis fulfilled in the wide sense [3]. However, channels in realisticenvironments, especially in high mobility scenarios illustratenon-stationary properties [4]-[7]. Authors in [8] reported thatthe WSS assumption can result in erroneous evaluation ofsystem performance. Therefore, for high mobility scenarios,

978-1-5386-4502-4/17/$31.00 ©2017 IEEE

the non-stationarity of channels in channel modeling has tobe considered. Most non-stationary channel models in theliterature were developed based on the movements of the MSand clusters. Authors in [9] proposed a non-stationary IMT-Advanced high-speed train (HST) channel model [10]. Time-varying parameters including delays, powers, and angles werecalculated according to the movement of the receiver andclusters. A three-dimensional (3-D) wideband non-stationarygeometry-based stochastic channel model (GBSM) for V2Vchannels was proposed in [11]. The model is composed ofa line-of-sight (LoS) component, a two-sphere model, and anelliptic-cylinder model. The last two models capture the effectsof moving vehicles and fixed roadside scatterers, respectively.The movements of the transmitter and receiver result in time-varying angles of departure (AoDs) and time-varying angles ofarrival (AoAs), which can be obtained based on the geometricconstruction.

A common assumption in the channel modeling is thatthe MS moves along a straight line with a constant speed.However, in real-world environments, the MS can move alongdifferent trajectories with time-varying speed. Relaxing theconstant velocity assumption can help to develop more realisticnon-stationary channel models. In [12], a non-stationary chan-nel model for isotropic scattering environments consideringvelocity variations of the MS was proposed. The temporalACF and Wigner-Ville spectrum of the proposed model werederived. In [13], a non-stationary mobile-to-mobile (M2M)channel model allowing for velocity variations of the MS wasproposed. The authors developed the model under isotropicscattering condition and illustrated that the correlation prop-erties of the M2M channel can be significantly affected bythe variations of the MS velocity. Further extension of [13]were reported in [14]. where the correlation properties of theproposed model were investigated under non-isotropic scatter-ing condition. However, models in [13]. [14] are single-inputsingle-output (SISO) M2M channel models and the modelsin [12]-[14] omitted the moving scatterers in propagationenvironments.

In this paper, a non-stationary MIMO channel model forstreet corner scenarios is developed using a geometry-based


approach. The proposed model takes into account both fixedand moving scatterers, which correspond to the roadsidebuildings and moving vehicles, respectively. The movementsof the MS and moving scatterers result in time-varying AoDsand AoAs, which make the model non-stationary. The contri-butions of this paper are summarized as follows.

(1) This paper proposes a non-stationary MIMO channelmodel for street corner scenarios. The velocity variations ofthe MS and scatterers are considered, which makes the modelmore general and realistic.

(2) From the proposed model, we derive the statistical prop-erties including spatial CCF, temporal ACF, and Wigner- Villespectrum, which are sufficiently general and can incorporatethe effects of velocity variations of the MS and scatterers.

The remainder of this paper is organized as follows. Sec-tion II describes the geometric corner scattering channel modeland gives the expression of the channel impulse response(CIR). Statistical properties of the model are studied in Sec-tion III. Section IV presents some numerical and simulationresults. Finally, conclusions are drawn in Section V.

II. A NEW GBSM FOR STREET CORNERSCENARIOS Fig. I: The geometric comer scattering model containing fixed and

moving scatterers.

where

(5)

(3)

(2)

i = 1,2

K pq . npq . e-·i ko·f;p'1 (t )K pq+ 1

t 1JR(t) T,<PLoS(t) = io -).._. cos(f3L~S (t) - rPR(t)) dt

h~q(t) =

In (2)~(4), K pq is the Ricean factor and npq is the totalpower of Tp~R q link. Here, 7]1, 7]2, and 7];3 designate howmuch the rays caused by fixed scatterers and moving scattererscontribute to the total scattered power npq/ (Kpq + 1), respec-tively, and meet the condition 7]1+ 7]2+ 7];3 = 1. ko = 27r/).. isthe wave number, where ).. is wavelength. The initial phases(}n l' (}n2' and (}na are independent and identically distributed(i.i.d.) random variables and uniform distributed from -7r to7r. The time-varying Doppler frequency components causedby the moving MS and s(na) have to be expressed as integralforms

hpq(t) = h~~s(t) + h~OS(t)

2

= h~~S(t) + L h~q(t) + h';};J(t) (1);=1

The narrowband MIMO geometric street corner scatteringchannel model is shown in Fig. 1. It is assumed that there areN 1 effective scatterers denoted by sen,) (n1 = 1, ... , N 1) lieon the buildings W1 located on the right side of this figure, andN 2 effective scatterers denoted by s(n 2 ) (n2 = 1, ... , N 2 ) lieon the buildings W2 located at the bottom of this figure. Themoving vehicles or pedestrians in the propagation environmentare modeled as N;3 effective scatterers which are denoted bys(na) (n;3 = 1, ... , N;3). We assume that the base station (BS)is fixed. The MS and s(na) can be moving with time-varyingvelocities, including time-varying speeds and time-varyingangles of motion (AoMs). The antenna array orientation of theMS is changed with the AoM of the MS. Both the BS and MSare equipped with uniform linear array with omnidirectionalantennas. It is worth to mention that only the LoS component,and single-bounced (SB) components are considered in thismodel. The key parameters of this model are summarized inTable I.

The MIMO fading channel can be expressed as a matrixH(t) = [hpq(t) ]lvITXlvIW The received complex fading enve-lope between the p-th (p = 1, ...!vIT ) transmit antenna elementT p and the q-th (q = 1, ... !vIR ) receive antenna element R q is asuperposition of the LoS component h~~s (t) and non-line-of-sight (NLoS) component h~S(t). The latter is composed ofSB rays caused by fixed scatterers h~q(t), and SB rays causedby moving scatterers h';};J (t). The CIR of the proposed modelis given by


TABLE I: Summary of key parameter definitions.

OT, OR antenna element spacings at the BS and MS, respectively'YT, 'YR(t) antenna tilt angles at the BS and MS, respectively

VR(t), VM(t) speeds of the MS and s\na), respectively<PR(t), <PM(t) AoMs of the MS and Ss">' : respectively

«n . ou accelerations of the MS and sena), respectivelyWR , WAJ angular speeds of the MS and Sena), respectively

OI.':;'iJ, OI.,:;'a)(t) (i = 1,2) AoDs of the waves impinging on S(n;J and S(n,,), respectivelyf3~'iJ (t) (i = 1,2,3) AoAs of the waves traveling from s(n;J

f3",;~ (t) AoA of the LoS pathI;pq(t), I;pn; (t), I;n;q(t), hT1(2)' distances d(Tp, Rq), d(Tp, seniJ), d(SeniJ, Rq),

and hR(M)1(2)(t) (i=I,2,3) d(OT, W 1(2) ), and d(OR(M), W 1(2) )

where to indicates initial time. Similarly, the time-varyingAoM of the MS (8(n a» can be expressed as

r VR(t) (n)<PF;(t) = io ->.- ocos(f3R ' (t) -¢R(t)) dt (i = 1,2) (6)

_ t VR(t) , (na)<PM l (t) - io ->.- 0 cos(f3R (t) - ¢R(t))dt (7)

_r VM(t) , (na)<PM2(t) - io ->.- 0 cos(aT (t) - ¢s(t))dt (8)

_ t VM(t) , (na)<PMa (t) - io ->.- 0 cos(f3R (t) - ¢s(t) )dt. (9)

The time-varying speed of the MS (8(na» can be calculatedas

¢R(M)(t) = ¢R(M)(tO) + wR(M) 0 t. (11)

In (2H4), epq(t) and epq,n; (t) = epn; (t) + en;q(t) aretime-varying travel distances of the waves through the linkTp - R q and Tp - 8(n;) - R q , respectively. Note thatmin{eTR,eTn;,en;R} » max{oT,oR}, the travel distancescan be calculated as

epq(t) ~ epR(t) - kqoRcos ("m(t) - f3[r1's(t)) (12)

epn,(2) ~ eTn,(2) - kpoTcos ('YT - a~"(2)) (13)

en'(2)q(t) ~ en'(2)R(t) - kqoRcos ('YR(t) - f3~l(2)(t))(14)

epna (t) ~ eTna (t) - kpoTcos ('YT - a¥:")(t)) (15)

en"q(t) ~ ~naR(t) - kqoRcos ('YR(t) - (3);'a) (t)) (16)

where kp = (MT - 2p + 1)/2 and k q = (MR - 2q + 1)/2,

~pR(t) ~ ~TR(t) + kpoTcos (f3[r1's(t) - 'YT) (17)

eTR(t) = V[hm(t) - hTl(t)]2 + [hm(t) - hT2(t)]2 (18)

~Tnl = hTl/ cos(a¥:'») (19)

(28)

(25)

(26)

(29)

(:3)() (hlvI1(t)-hTl(t))J-tT t = arctan hT2(t) - hlvI2(t)

(:3)( ) (hM, 2(t) - hm(t))J-tR t = arctan .hm(t) - hlvI1(t)III. STATISTICAL PROPERTIES

where

The moving cluster 8(n,,) is depicted as a disc with a certainangular spread distributed in the horizontal plane. The rela-tionship between a¥:a)(t) and (3);'a)(t) can be determined as

(na) (tan (a¥:a)(t) - J-t~)(t)) 0 eTna(t)) (:3)f3R (t) = arctan ~naR(t) +J-tR (t)

(27)

A. Spatial-Temporal Correlation FunctionThe spatial-temporal correlation function of CIR [IS] can

be defined as

E{h;'q,(t - ~t)hpq(t + ~t)}Ph(t, OT, OR, t:J.t) = (30)ylo.P'q,o.pq

where E{-} is expectation operator and (0)* denotes the com-plex conjugate operation. Since the clusters in the proposed

~n2R(t) = hm(t)/sin(-f3);',) (t)) (22)

eTna(t) = V[hlvI1(t) - hTl(t)]2 + [hlvn(t) - hT2(t)]2~----,------,--...,,-------,--o;-;;----;o-:-----,------,------,-------,--~( 23)

~naR(t) = V[hM1(t) - hm(t)]2 + [hlvI2(t) - hm(t)]2.(24)

Based on the geometric construction, a¥:',) is in the rangeof (-1r/2, 0), and a¥:2) is in the range of (-1r, -1r/2). For SBrays, the AoDs and the AoAs are interdependent. The AoAscan be calculated based on the AoDs and the movements ofthe MS and 8(na) 0 For fixed clusters, the relationship betweena(n,) and (3);',) (t) and the relationship between a¥:2) andf3~2) (t) can be calculated as

f3 (n,) ( ) (hT2 - hm(t) +, tan(a¥:,)) 0 hTl)R t =arctan ()h-n: t

f3 (n2)( ) _ (hTl - hm(t) +, cot(a¥:2») 0 hT2)R t - arccot ( ) .hm t

(20)

(21)

~nlR(t) = hm(t)/cos(f3);',) (t))

~Tn2 = hTd sin(_a¥:2»)


2

Ph(t, DT, DR, !:it) =pkOS(t, DT, DR, !:it) + L P~' (t, DT, DR, !:it)i=l

B. Spatial CCFBy substituting (1) to (31) and imposing !:it = 0, the spatial

CCF can be obtained as

models are assumed to be independent of each other, thespatial-temporal correlation function can be further expressedas the summation of the LoS component, fixed cluster com-ponent, and moving cluster component:

(39)• ( (17.,,) t) d (17.,,)paT " aT .

'T/:3 j1r eiko(~P'1,n,,(t-~t)-~P'1,n,,(t+~t»)K pq + 1 -1r· ei21r( \PM l (t+ ~t )-\PM, (t- ~t»)

· e-·i21r(\PM2(t+,¥-)-\PM2(t-~t»)

· e-.i21r( \PM" (t+ '¥-) -\PM" (t- ~t ) )

p~(t, !:it)

For the SB component due to moving scatterers

D. Wigner-Ville Spectrum

The Wigner-Ville spectrum [16] of the complex CIR iscalculated through the Fourier transform of local temporalACF in terms of !:it, and can be calculated as

{CO }_ * !:it !:it -i21rf!:>.tSh(t, f) -E -L h (t - 2""" )h(t + 2""") . e d!:it

(32)

(31)

2

Ph(t, DT, DR) =PkOS(t,DT,DR) + LP~'(t,DT,DR)i =l

+ /(;(t, DT, DR)'For the spatial CCF of the LoS component,

where P = (pi - p)DT/>', Q = (ql - q)DR/>'. For the spatialCCF of the NLoS component,

2

p~oS(t, DT, DR) = LP~' (t, DT, DR) + p~(t, DT, DR) (34)i=l

For the LoS component,

pkOS(t, !:it) K pq eiko(~'1P(t-~t)-~'1P(t+,¥-»)K pq + 1. ei21r( \PLoS(t+ ~t )-\PLoS(t- ~t»). (37)

For the SB component associated with fixed scatterers

p~'(t,!:it) = 'T/i j1r eiko(~n,R(t-~t)-~n,R(t+,¥-»)K pq + 1 -1r

. ei21r(\PF,(t+,¥-)-\PF,(t-~t»)p(a~'») da~') (i = 1,2).(38)

(40)

IV. RESULTS AND ANALYSIS

co

= JPh(t,!:it)· e-·i21r f !:>.t d!:it .-00

In this section, numerical and simulation results of statistics,i.e., spatial CCF, temporal ACF, and Wigner- Ville spectrum arepresented. The AoDs and AoAs are assumed to follow vonMises distribution [17]. The PDF of von Mises distributioncan be expressed as

eKcos ("'-/-,)

f(x I P"K,) = 27rlo(K,) (41)

where x E [-7r,7r), 10 ( , ) denotes the zero order modifiedBessel function of the first kind, p, is the mean value of angles,and 11, controls the angular spread. The discrete values ofangular parameters are obtained using the modified method ofequal areas (MMEA) approach [18] and 50 cisoids are adoptedin the simulation model. We assume that there are threeclusters in the propagation environment. Two fixed clusterson the roadside buildings are located around transmitter andreceiver, respectively, which contribute most to the receivedpower. The third cluster is a moving cluster results fromvehicles or pedestrians. The simulation parameters are selectedas follows: f = 2 GHz, M T = 2, M R = 2, "IT = 7r/2,'YR(tO) = 7r/2, VR(tO) = 5 m/s, cPR(tO) = 0, VM(tO) = 5 m/s,cPM(tO) = 0, o-u = 0.2 m/s", WM = 7r/20 8-1, h-r: = 8 m,hT2 = 70 m, hm(to) = 60 m, hm(to) = 4 m, hs1(to) =80 m, hS2 (tO) = 40 m, p,(n,) = -7r/4, p,(n 2 ) = -57r/9,11,(17.,) = 10 (i = 1,2,3), Kp,q = 0, Kp',q' = 0, 'T/1 = 0.4,'T/2 = 0.5, 'T/:3 = 0.1.

The absolute values of the time-varying spatial CCF areshown in Fig. 2. The spatial CCF varying with time t andcaused by the velocity variations of the MS and the movingscatterers can be observed. The results in Fig. 3 are obtainedwhen the MS travels along a straight line with an angularspeed. The differences between the two curves stem from thetime-varying antenna orientation of the MS. Besides, (33) and

(33)

(Kpq + l)(Kp'qp+ 1). eiko{ q.cos(,9~~~(t)-'YR(t»)-P.cos(,9~~~(t)-'YT)}

and can be calculated as

PNLos(t " ") _ 'T/ih ,UT,UR-y(Kpq + l)(Kp'q' + 1)

j 1r .iko {P.cos (",~L'\t)-'YT) +q.cos (,9~L' ) (t)-'YR(t») }e-1r

( (17.,») d (17.,) (. 1 2 3) (35)'paT aT l="

where p ( a~'») is the probability density function (PDF) of(17.,)aT .

C. Temporal ACFThe temporal ACF is calculated by substituting (1) to (31)

and imposing DT = 0 and DR = 0, and can be expressed as2

Ph(t,!:it) = pkOS(t,!:it) + LP~'(t,!:it) + p~(t,!:it). (36)i =l


Antenna spacing, OR/'\ 10 a Time, t (8) Time delay, 6.t (8) Time, t (8)

Fig. 2: The absolute value of the local spatial CCF of the proposedmodel (VR(tO) = 5 mis, <PR(tO) = 0, aR=0.2 m/s", WR = 1r/IO 8-

1,

VM(tO) = 5 mis, <pM(tO) = 0, o.u = 0.2 m/s", WM = 1r/20 8-1).

Fig. 4: The absolute valueof the local temporal ACF of the proposedmodel (VR(tO) = 5 mis, <PR(tO) = 0, aR=0.2 m/s", WR = 1r/IO 8-

1,

VM(tO) = 5 mis, <pM(tO) = 0, o.u = 0.2 m/s', WM = 1r/20 8-1).

0.9r...u 0.8U::$ 0.7~,~O. 6

'0~ 0.5

~ 0. 4

'"-<;::: 0.3"2'~ 0 2~ .

0. 1

II,

I,,, '""",\

- WR

= 0 5-1, theoretical

o (DR = 0 S-1, simulation

- (DR = rr./1 0 S-1, theoretical

(DR = n:/10 S-1, simulation

2 4 6l\,fS antenna spacing, OR/'\

0.9r...S2 0.8

'" 0.71)[0.6'"~ 0.5

~ 0. 4

'"~ 0.3o'~ 0 2~ .

0. 1

Ir-,,rI

,•,,,

0 . 01

- WR

= 0 5-1, theoretical

o (DR = 0 S-1, simulation

- - - WR

= n/10 5-1, theoretical

1:1 (DR = n/10 5-1, simulation

// - \

, I, \, - I

I _ PI

\ I

'..'

0.02 0.03 0. 0 4Time delay, 6.t (8)

Fig. 3: The absolute value of the local spatial CCF of the proposedmodel at different angular speeds (v R (to) = 5 mis, <PR (to) = 0,an = 0.2 m/s", VM(tO) = 5 mis, <PM(tO) = 0, as: = 0.2 m/s",WM =1r/20 8-

1).

(35) indicate the acceleration have no influence on the spatialCCF.

Fig. 4 shows the absolute value of the time-varying temporalACF of the proposed model. Both the movements of clustersand velocity variations (including acceleration and angularspeed) result in time-varying ACF. Results of Fig. 5 and Fig. 6show the influences of angular speed and acceleration of theMS on the temporal ACFs. We can find that both the angularspeed and acceleration result in weak temporal correlation.Moreover, we find that the angular speed has a greater impacton the temporal correlation than that of the acceleration.

The Wigner-Ville spectrums of the proposed model atdifferent angular speeds and accelerations are shown in Fig. 7.Compared with Fig. 7 (a), which is obtained when the MSmoves along a straight line with a constant speed, Wigner-Ville spectrum varying over time due to the velocity variations

Fig. 5: The absolute valueof the local temporal ACF of the proposedmodel at different angular speeds (vR ( to) = 5 mis, <PR ( to) = 0,an = 0.2 m/s', VM(tO) = 5 mis, <pM(tO) = 0, o.u = 0.2 m/s',WM = 1r/20 8-

1).

of the MS can be observed. The time-varying Wigner- Villespectrum in Fig. 7 (b) is caused by the angular speed of theMS. In Fig. 7 (c), the speed of the MS increases when the MSmoves with an acceleration, which results in increased valuesof Doppler frequency shifts. Fig. 7 (d) shows the combinedeffects of angular speed and acceleration of the MS on theWigner- Ville spectrum of the proposed model.

V. CONCLUSIONS

In this paper, we have proposed a non-stationary MIMOchannel model for street comer scenarios. The model wastaken into account both fixed and moving scatterers. Thespeeds and trajectories of the MS and moving scatterers areallowed to change. All of these make the proposed model moregeneral and flexible. Statistical properties including spatialCCF, temporal ACF, and Wigner- Ville spectrum have been


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[11]

[18]

[17]

0.04

--aR = 0 m/s2 , theoretical

o aR

= 0 m/s2 , simulation

- - - aR = 0.2 m/s2, theoretical

A aR

= 0.2 m/s2, simulation\\,\,,¥

-100 0 100Frequency. f (Hz)

(b) an = 0 m/s", WR = 7f/10 S-l[12]

0.02 0.03Time delay, 6.t (s)

0.01

0.2

0. 1

-100 0 100Frequency. f (Hz)

(a) an = 0 m/s", WR = 0 S-l

0 .4

1.5 1.5 [13]~ ~- -i 1 i 1.5 .5C< C< [14]

0.5 0.5

1.5 1.5

~ Z- 1 -i.~

1.5 [10]C< C<

0.5 0.5

'0 0 .5

~

~

""§ 0 7~ .

~ 0.6

~ 0.3

~<

Fig. 6: The absolute value of the local temporal ACF of the proposedmodel at different accelerations (vR ( to) = 5 mis, rPR ( to) = 0, W R =7f/10 8-

1, VM(tO) = 5 mis, rPM(tO) = 0, as: = 0.2 m/s", WM =7f/20 8- 1).

derived. Numerical and simulation results have been presentedand analyzed under the non-isotropic scattering condition.We have found that the velocity variations of the MS andmoving scatterers can have significant influences on channelstatistics and exacerbate the non-stationarity of the channels.The proposed model provides a fundamental framework forfuture non-stationary channel modeling.

ACKNOWLEDGEMENT

The authors grateful!y acknowledge the support from the NaturalScience Foundation of China (Grant No. 61371110), Fundamen-tal Research Funds of Shandong University (2017JC009), EPSRCTOUCAN project (Grant No. EPIL020009/1), and EU H2020 RISETESTBED project (Grant No. 734325).

o-100Ftequen(g., f (Hz) 100 -100Frequen(g., f (Hz) 100 [15]

(c) o.« = 1 m/s', WR = 0 S-l (d) o.« = 1 m/s', WR = 7f/10 S-l

Fig. 7: Wigner-Ville spectrum of the proposed model at differentangular speeds and accelerations (vR (to) = 5 mis, rPR (to) = 0,VM(tO) = 5 mis, rPM(tO) = 0, o.u = 0.2 m/s", WM = 7f/20 8-

1). [16]

a non-stationary mimo channel model ... - southeast university

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