An Efficient V2X Based Vehicle Localization Using Single RSU … · Received March 15, 2019,...

An Efficient V2X Based Vehicle Localization Using Single RSU andSingle Receiver

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Citation for the original published paper (version of record):Ma, S., Wen, F., Zhao, X. et al (2019)An Efficient V2X Based Vehicle Localization Using Single RSU and Single ReceiverIEEE Access, 7: 46114-46121http://dx.doi.org/10.1109/ACCESS.2019.2909796

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Received March 15, 2019, accepted April 3, 2019, date of publication April 10, 2019, date of current version April 17, 2019.

Digital Object Identifier 10.1109/ACCESS.2019.2909796

An Efficient V2X Based Vehicle LocalizationUsing Single RSU and Single ReceiverSUGANG MA1, FUXI WEN 2, (Member, IEEE), XIANGMO ZHAO1,ZHONG-MIN WANG3, AND DIANGE YANG 41School of Information Engineering, Chang’an University, Xi’an 710064, China2Department of Electrical Engineering, Chalmers University of Technology, 412 96 Gothenburg, Sweden3School of Computer Science and Technology, Xi’an University of Posts and Telecommunications, Xi’an 710121, China4State Key Laboratory of Automotive Safety and Energy, Department of Automotive Engineering, Tsinghua University, Beijing 100084, China

Corresponding authors: Xiangmo Zhao ([email protected]) and Diange Yang ([email protected])

This work was supported in part by the International Science and Technology Cooperation Program of China under Contract2016YFE0102200, in part by the State Key Laboratory of Automotive Safety and Energy under Project KF1804, in part by the EuropeanCommission H2020-MSCA-IF-2015 under Project 700044, and in part by the Science and Technology Innovation Project of ShaanxiProvince under Grant 2016KTZDGY04-01.

ABSTRACT High accuracy vehicle localization information is critical for intelligent transportation systemsand future autonomous vehicles. It is challenging to achieve the required centimeter-level localizationaccuracy, especially in urban or global navigation satellite system denied environments. Here we proposea vehicle-to-infrastructure (V2I)-based vehicle localization algorithm. First, it is low-cost and hardwarerequirements are simplified, the minimum requirement is a single roadside unit and single on-boardreceiver. Second, it is computationally efficient, the available V2I information is formulated as anover-determined system. Then, the vehicle position is estimated in a closed-form manner via the widelyused weighted linear least squares (WLLS) method and meter level accuracy is achievable. Furthermore,the numerical performance of WLLS is consistent with the theoretical results in larger signal-to-noise ratioregion.

INDEX TERMS Vehicle localization, vehicle-to-everything (V2X), vehicle-to-infrastructure (V2I), roadsideunit (RSU), linear least squares.

I. INTRODUCTIONAs sub-classes of Vehicular ad hoc networks (VANET),intelligent transport systems and connected vehicles requireprecise and real time vehicle positions, which creates therequirements for efficient and accurate vehicle localiza-tion [1]. Global navigation satellite system (GNSS) is oneof the most commonly used vehicle localization techniques.However, it suffers from poor reliability, especially in urbanenvironments [2]. Although real-time kinematic method canachieve centimeter level accuracy, the signal availabilityremains a problem. On the other hand, signal availabilitycould be improved by combiningmultiple satellite navigationsystems to increase the number of available satellites [3].Furthermore, GNSS is often integrated with inertial mea-surement unit (IMU) to design a hybrid localization sys-tem. This IMU information can be used for moving vehicleself-localization. The vehicle position relative to its initial

The associate editor coordinating the review of this manuscript andapproving it for publication was Venkata Ratnam Devanaboyina.

position can be estimated by dead reckoning. However, it suf-fers from the accumulated errors as the vehicle moves [2].

For the future autonomous driving applications, bothintegrity and localization accuracy must be improved [4].As a key component for the future intelligent connected vehi-cles, the vehicle-to-everything (V2X) services are standard-ized by the 3rd generation partnership project (3GPP) [5].The V2X information from connected vehicles and fixedroadside infrastructures can be integrated to the existinglocalization systems cooperatively to improve both accuracyand robustness at a relatively low cost, sensing range limita-tions associated with on-board sensors are also addressed [6].We first review some of the widely used V2X localiza-tion techniques: vehicle-to-vehicle (V2V), vehicle-to-feature(V2F) and vehicle-to-infrastructure (V2I).

The V2V localization techniques integrate informationfrom adjacent connected vehicles with on-board sensingcapabilities. These vehicles can be considered as virtualanchor nodes. However, these methods are GNSS dependent

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VOLUME 7, 2019

https://orcid.org/0000-0003-3686-1446https://orcid.org/0000-0003-0825-5609

S. Ma et al.: Efficient V2X-Based Vehicle Localization Using Single RSU and Single Receiver

and inter-vehicle range/angle measurements are required aswell to apply the trilateration or triangulation techniques.However, the inter-vehicle measurements are sensitive to theshadowing effects either from other vehicles or from the bodyof the car itself.

Recently, V2F based autonomous vehicle’s self-localization become more popular due to the availabilityof locally and globally dynamic high definition (HD) mapand development of light detection and ranging (LiDAR)technologies [7]. The detected features are used as refer-ence anchors for localization. While the main challengingis processing and transmission the high data volume withlow latency [8], [9]. On the other hand, an offline map isnot required if the non-cooperative features can be detectedand jointly localized by the connected vehicles [10]. How-ever, perfect association between vehicle measurements andsensed features is required.

Besides HD maps, roadside infrastructures also provideuseful vehicle position information. Base station and roadsideunit (RSU) are widely used facilities for V2I communica-tions. 5G technology presents a new paradigm to provideconnectivity and high data-rate services to vehicles. Usingthe existing communication hardware, it also provides oppor-tunities for accurate vehicle localization from a single 5Gbase station [11]. To achieve this, large bandwidth combinedwith large scale antennas are required at both the base stationand vehicle sides [12], [13]. RSU is another GNSS-free andlow-cost infrastructure for localization. Each vehicle esti-mates its position by extracting the position related informa-tion from the radio signals transmitted by the nearby RSUswith known position [14].

RELATED WORK AND CONTRIBUTIONSHere we focus on vehicle-to-RSU based localization. Toreduce deploying costs, RSUs are often sparsely dis-tributed and limit the application of the trilateration andtriangulation-based techniques, which requires multipleRSUs to obtain location estimates. Consequently, single RSUbased localization techniques have been developed recentlyin the literature.

An inertial navigation system (INS)-assisted and sin-gle RSU based vehicle localization framework is proposedin [15]. While two types of RSU at the entry points and themiddle of the road are needed to determine the vehicle drivingdirection. Recently, a single RSU based localization approachusing angle-of-arrival and range information between vehicleand RSU is proposed in [16]. However, multiple calibratedreceivers either from the RSU or vehicle side are requiredto obtain an accurate angle information. Other positioningtechniques are proposed in [17] and [18] by exploiting theangle information between vehicle and RSU, in conjunctionwith velocity vector and the broadcast RSU position. Againmultiple receivers are required to estimate the angle informa-tion from the received radio signals.

The constraint of using single RSU and single on-boardreceiver for vehicle localization poses a significant challenge

TABLE 1. Brief comparison of the existing RSU-based localizationalgorithms.

and limits the applications of the existing RSU-based local-ization techniques. In this paper, an IMU-assisted single RSUand single on-board receiver-based localization algorithm isproposed to eliminate this challenge.

Our contributions are summarized as follows:• Low-Cost and Low Complexity:Hardware requirements are simplified, the minimumrequirement is a single roadside unit (RSU) and singleon-board receiver.

• Computationally Efficient:The available vehicle-to-RSU information is reformu-lated as an over-determined system, which can be solvedin a closed-form manner by the widely used linear leastsquares (LLS) or weighted LLS (WLLS) methods.

• Theoretical Analysis:The theoretical analysis for WLLS are carried out, if theerror of extracted range information is Gaussian dis-tributed with zero means. Furthermore, the theoreticalroot means square position error (RMSE) performanceis provided.

The rest of the paper is organized as follows. In Section II,the problem is formulated. In Section III, we derive the pro-posed LLS and WLLS estimators-based vehicle localizationapproaches. The RMSE of the proposed algorithm is analyzedin Section IV. Simulation results are provided in Section V toevaluate the localization accuracy of the proposed localiza-tion algorithm. And conclusions are drawn in Section VI.

II. PROBLEM FORMULATIONLocalization in wireless sensor networks is the process offinding a target node’s absolute position using single or mul-tiple anchor nodes. The positions of the anchor nodes areknown [19]. As shown in Fig. 1, trilateration and triangula-tion are the widely techniques for radio-based localization byexploiting range and/or angle information between the anchorand target nodes [19]. Received signal strength (RSS) time-of-arrival (TOA), time difference-of-arrival (TDOA) andangle-of-arrival (AOA) of the emitted signals are commonlyused measurements for radio-based location [20]. Basically,TDOA requires multiple synchronized anchor nodes andAOA requires multiple calibrated receivers on the targetnodes [20].

In this paper, we consider the vehicle localization problemby exploiting V2I information. The vehicle trajectory can bearbitrary and within the communication range of a RSU withfixed position p = [x y]T for 2D scenarios. The positioninformation and ID of RSU are broadcast to the vehicle.

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FIGURE 1. An illustration of 2D localization (a) trilateration,(b) triangulation, (c) range and angle (hybrid).

FIGURE 2. Single roadside unit and single receiver based GNSS-freevehicle localization. Legend: Variables in BLUE and RED colors denote theavailable noisy measurements and the unknown parameters to beestimated, respectively.

We assume that the vehicle velocity remains constant duringvery short time intervals. Let vk+1 = [vx,k+1 vy,k+1]T be thevector velocity during t ∈ (tk , tk+1], which can be obtainedfrom on board sensors at time tk [21]. As shown in Fig. 2,pk = [xk yk ]

T denotes the vehicle position at time tk .The processing of localization is triggered at time t0 andp0 = [x0 y0]

T .Remark 1: Even though 2D localization problem is dis-

cussed in detail, extension to 3D scenarios are straightfor-ward by setting RSU position p = [x y z]T , vehicle velocityvector vk+1 = [vx,k+1 vy,k+1 vz,k+1]T , and the unknownvehicle position pk = [xk yk zk ]

T .For the proposed two-step method, we first formulated the

available vehicle-to-RSU information as an over-determinedsystem. Then the vehicle position is estimated in aclosed-form manner via the widely used linear least squaresmethod.

III. PROPOSED LOCALIZATION ALGORITHMLet dk be the range information between the vehicleand RSU at time tk . After obtaining the vector velocityvk = [vx,k vy,k ]T , unknown vehicle position pk can be

described as

pk = pk−1 + vkτk ⇒

{xk = xk−1 + vx,jτkyk = yk−1 + vy,jτk

, (1)

where the kth time interval τk = tk − tk−1.It is straightforward to adopt the following kinematic

model from (1),{xk = x0 +

∑kj=1 vx,jτj = x0 +1xk

yk = y0 +∑k

j=1 vy,jτj = y0 +1yk, (2)

where k is the number of measurements can be used forvehicle localization, the corresponding accumulated rangeinformation is defined as,

1xk =k∑j=1

vx,jτj and 1yk =k∑j=1

vy,jτj. (3)

Remark 2: Assuming that from the trigger point, vehiclevelocity vectors vk and τk are known up to k. After obtainingtrigger position, vehicle positions {p1, p2, · · · , pk} can beinferred from p0 via (2). The unknown parameters to beestimated is p0 = [x0 y0]

T .The range dk between RSU and vehicle is

(xk − x)2 + (yk − y)2 = d2k . (4)

Substituting (2) into (4) yields

(x0 +1xk − x)2 + (y0 +1yk − y)2 = d2k . (5)

Let

R0 = x20 + y20 and R = x

2+ y2, (6)

we expand (5) to obtain

2(1xk − x

)x0 + 2

(1yk − y

)y0 + R0

= d2k − (1xk − x)2− (1yk − y)2. (7)

Equation (7) can be rewritten as

Aθ = b, (8)

where

A =

2(1x1 − x) 2(1y1 − y) 1...

2(1xj − x) 2(1yj − y) 1...

......

2(1xk − x) 2(1yk − y) 1

=

a1...

aj...

ak

(9)

θ =[x0 y0 R0

]T=[x0 y0 x20 + y

20

]T (10)

b =

d21 + Q1−R...

d2j + Qj − R...

d2k + Qk − R

=

b1...

bj...

bk

(11)

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and for j = 1, 2, · · · , k ,

Qj = 1xj(2x −1xj

)+1yj(2y−1yj). (12)

That is, A is known, θ contains the unknown parametersand b is the observation vector.Remark 3: Now we extend the localization algorithm to

3D scenarios. Let pk = [xk yk zk ]T be the 3D vehicle position.

The k-th row of amended measurement matrix A (9) andparameter θ (10) are defined as

ak =[2(1xk − x) 2(1yk − y) 2(1zk − z) 1

](13)

and

θ =[x0 y0 z0 R0

]T (14)where R0 = x20 + y

20 + z

20. Meanwhile, R and Qk in (11) are

substituted with

R = x2 + y2 + z2, (15)

and

Qk = 1xk(2x −1xk

)+1yk (2y−1yk )+1zk (2z−1zk ).

(16)

A. LINEAR LEAST SQUARES (LLS)In practice, dj is substituted by its biased estimate d̂j and isdescribed as

d̂j = dj + ej, j = 1, 2, · · · , k, (17)

where

ej ∼ N (µj, σ 2j )+N (0, σ2), (18)

is the error component. Here, µj and σ 2j are the mean andvariance of range uncertainty, and σ 2 denotes the variance ofthe white noise.

Substitute b in (8) with b̃, we have Aθ ≈ b̃. The LLSestimate of θ is obtained by minimizing:

J (θ̃ ) =(Aθ̃ − b̃

)T (Aθ̃ − b̃

), (19)

where θ̃ is the variable for θ . The closed form solution isgiven by

θ̂ =(ATA

)−1AT b̃. (20)

The estimated trigger point is

p̂0 =[x̂0 ŷ0

]T=[θ̂1 θ̂2

]T. (21)

and vehicle positions p̂j, {j = 1, 2, · · · , k} can be inferredfrom (2).

The proposed LLS-based vehicle localization algorithm issummarized in Algorithm 1.

Algorithm 1 LLS-Based LocalizationInput: p, tj, dj and vj, for j = 1, 2, · · · , k .Output: Estimates p̂0 and p̂j, for j = 1, 2, · · · , k .1: for j = 1 to k do2: Calculate 1xj and 1yj by (3)3: end for4: Construct A by (9)5: for j = 1 to k do6: Calculate Qj and R by (12) and (6), respectively.7: end for8: Construct b by (11)9: Estimate θ using LLS (20).10: Estimate p̂0 by (21) and p̂j, j = 1, 2, · · · , k , by (2).

B. WEIGHTED LLSEmploying the technique proposed in [22], the LLS algo-rithm can be improved by including a second WLLS step byexploiting the constraint (6). Assuming that θ̂1 and θ̂2 of (20)is sufficiently close to x0 and y0, then we have{

θ̂21 − x20 ≈2x0(θ̂1 − x0)

θ̂22 − y20 ≈2y0(θ̂2 − y0)

(22)

Based on (6) and with the use of (22), we construct

η = Dz+ r, (23)

where

D =

1 00 11 1

, (24)z =

[x20 y

20

]T, (25)

η =[θ̂21 θ̂

22 θ̂3

]T, (26)

r =[2 x0(θ̂1 − x0) 2 y0(θ̂2 − y0) θ̂3 − R0

]T(27)

Note that z is the unknown parameter to be estimated and thecovariance of r is

Cr =

2x0 2y01

Cθ̂

2x0 2y01

. (28)where C

θ̂is given in (42). In practice, since x0 and y0 in (28)

are unknown, they are substituted with θ̂1 and θ̂2, respectively.The WLLS solution of z is

ẑ =(DTC−1r D

)−1DTC−1r η. (29)

As there is no Since the sign information for x0 and y0 can-not be recovered from z, the improved position estimate p̂0,is determined as

p̂0 =[sgn(θ̂1)

√ẑ1 sgn(θ̂2)

√z2]T. (30)

where sgn represents the sign function. Again the vehiclepositions p̂j, {j = 1, 2, · · · , k} can be inferred from (2). Theproposed method is summarized in Algorithm 2.

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Algorithm 2 Weighted LLS-Based LocalizationInput: p, tj, dj and vj, for j = 1, 2, · · · , k .Output: Refined estimates p̂0 and p̂j, for j = 1, 2, · · · , k .1: Obtain θ̂ from Algorithm 1.2: Construct h = Gz+ r from (23)-(27).3: Calculate covariance matrix Cr by (28).4: Estimate z using WLLS (29).5: Obtain the refined p̂0 by (30).6: Estimate the refined p̂j for j = 1, 2, · · · , k , by (2).

IV. PERFORMANCE ANALYSISCompared with Qj in (11), the dominant errors are from theestimated range d̂j. After ignoring the error in Qj, we have

b̃ = b+ w =[b̃1, · · · , b̃j, · · · , b̃k

]T, (31)

where observation error

w = [e21 + 2e1d1, · · · , e2j + 2ejdj, · · · , e

2k + 2ekdk ]

T . (32)

Lemma 1: The mean of b̃ is

E[b̃] = b+

σ 21 + σ2+ µ21 + 2µ1 d1...

σ 2j + σ2+ µ2j + 2µjdj...

σ 2k + σ2+ µ2k + 2µkdk

, (33)

where E[·] is the expectation operator. The covariance matrixof b̃ is given by

Cb̃ =

c1. . .

cj. . .

ck

, (34)

where

cj = 2(σ 4j + σ

4)+ 4d2j

(σ 2j + σ

2), j = 1, 2, · · · , k.(35)

Please refer to the proof provided in Appendix for moredetails.

A. BIAS AND MEAN SQUARE ERROR ANALYSISThe WLLS estimate of θ is given by

θ̂ = argminθ̃

J (θ̃ ), (36)

where

J (θ̃ ) =(Aθ̃ − b̃

)TC−1b̃

(Aθ̃ − b̃

), (37)

and weighting matrix C−1b̃

is given in (34). Equation (36)implies that

∇(J (θ̂ )

)=∂J (θ̃ )

∂ θ̃

∣∣∣∣θ̃=θ̂

= 0, (38)

where ∇(J (θ̂ )

)denotes the gradient vector evaluated at the

estimated value. If the estimation error θ̂ − θ is suffi-ciently small, take first-order Taylor series expansion of (38)around θ , we have

∇(J (θ̂ )

)≈ ∇

(J (θ )

)+H

(J (θ )

)(ˆθ − θ), (39)

where ∇(J (θ )

)and H

(J (θ )

)are the Hessian matrix and gra-

dient vector evaluated at θ , respectively. Bias is obtained bytaking the expected value on (39) [23],

bias(θ̂ ) ≈ −[E{H(J (θ )

)}]−1E{∇(J (θ )

)}. (40)

Similarly, the covariance matrix [22] is

Cθ̂≈

[E{H(J (θ )

)}]−1E{∇(J (θ )∇T

(J (θ )

)}[E{H(J (θ )

)}]−1.

(41)

Apply the bias (40) and MSE (41) formulas, we obtain:E{θ} = θ̂ and

Cθ̂=

(ATC−1

b̃A)−1

. (42)

For [x0, y0]T the RMSE is given by

RMSE =√[C

θ̂]1,1 + [Cθ̂ ]2,2, (43)

where [ ]i,j denotes the (i, j) entry of a matrix.

B. CRAMÉR-RAO LOWER BOUND (CRLB)The CRLB of p̂0 is analyzed as follows. Implicitly, p̂0 cor-responds to minimizing (37) subject to (6). Employing (5)and (34), we can write

p̂0 = argminp̃0

J (p̃0), (44)

where

J (p̃0) =K∑k=1

[(x̃0 +1xk − x

)2+(ỹ0 +1yk − y

)2− d̂2k

]22σ 4k + 4d

2k σ

2k

(45)

The Hessian matrix for J (p̃0) is expressed as

∂2 J (p̃0)

∂ p̃0∂ p̃T0

=

∂2 J (p̃0)∂ x̃20 ∂2 J (p̃0)∂ x̃0∂T ỹ0

∂2 J (p̃0)∂ ỹ0∂T x̃0

ãĂĂ ∂2 J (p̃0)∂ ỹ20

=[Jxx JxyJxy Jyy]. (46)

We start with

∂J (p̃0)∂ x̃0

=

K∑k=1

2(x̃0 +1xk − x

)2(x̃0 +1xk−x)σ 4k + 2d

2k σ

2k

+

K∑k=1

2(ỹ0 +1yk − y

)2(x̃0 +1xk−x)σ 4k + 2d

2k σ

2k

−

K∑k=1

2d̂2k (x̃0 +1xk−x)

σ 4k + 2d2k σ

2k

. (47)

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Using (47), we obtain

Jxx =K∑k=1

2[3(x̃0 +1xk−x)2 + (ỹ0 +1yk−y)2 − d̂2k

]σ 4k + 2d

2k σ

2k

,

(48)

Jyy =K∑k=1

2[3(ỹ0 +1yk−y)2 + (x̃0 +1xk−x)2 − d̂2k

]σ 4k + 2d

2k σ

2k

(49)

and

Jxy = Jyx =K∑k=1

4(x̃0 +1xk − x)(ỹ0 +1yk − y)

σ 4k + 2d2k σ

2k

. (50)

Evaluating these partial derivatives at p̃0 = p0 and employ-ing E{d̂2k } = d

2k yields

ηxx = E {Jxx} =K∑k=1

4(x0 +1xk − x)2

σ 4k + 2d2k σ

2k

, (51)

and

ηyy = E{Jyy}=

K∑k=1

4(y0 +1yk − y)2

σ 4k + 2d2k σ

2k

. (52)

Similarly, we get

ηxy = ηyx = E{Jxy}= E

{Jyx}

=

K∑k=1

4(x0 +1xk − x)(y0 +1yk − y)

σ 4k + 2d2k σ

2k

. (53)

Using (51), (52) and ((53)), we have

E

{∂2 J (p̃0)

∂ p̃0∂ p̃T0

} ∣∣∣p̃0=p0

=

[ηxx ηxy

ηyxãĂĂ ηyy

]. (54)

V. SIMULATION RESULTSSimulations are implemented to evaluate the WLLS (30) andLLS (21) vehicle localization algorithms, as a benchmarktheoretical RMSE (43) and CRLB (54) are also included.The position of RSU is p = [200 0]T and po = [1 2]

T .Sampling frequencies fs = 10 Hz and observation period 30 sare considered. The signal-to-noise ratio (SNR) in dB scale isdefined as

SNR = 10 log10(d2j /σ

2), (55)

where σ 2 is the variance of the Gaussian noise. Two sce-narios with accurate and inaccurate range information dj areconsidered.• Accurate range information, the error component in (17)is described as ej ∼ N (0, σ 2).

• Inaccurate range information, the error component isej ∼ N (µj, σ 2j )+N (0, σ 2).

All the results are obtained by averaging over 1000 inde-pendent runs.

1) ACCURATE V2I RANGE INFORMATIONIn the first test, constant velocity vector is considered andvk = [vx,k vy,k ]T = [6 4]T . Fig. 3 shows the RMSE ofthe LLS and WLLS methods versus SNR. The theoreticalvariances of the position estimates of the proposed estima-tor and CRLB are also included. WLLS-based localizationoutperforms LLS-based in terms of RMSE and meter levelaccuracy is achievable. Furthermore, the RMSEs of WLLSagree with (42) and approach the CRLB when SNR is largerthan 20 dB. As shown in Fig. 4, localization accuracy isfurther improved by increasing To and fs, while trade-offsshould be considered between computational complexity andlocalization accuracy.

FIGURE 3. RMSE versus SNR, observation time To = 30 s and samplingfrequency fs = 10 Hz.

FIGURE 4. RMSE versus observation time To, SNR is 30 dB and differentsampling frequency fs.

2) INACCURATE V2I RANGE INFORMATIONIn the second test, inaccurate V2I range information is con-sidered and vk = [vx,k vy,k ]T = [6 8]T , observation timeTo = 30 s and sampling frequency fs = 10Hz. As shown

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FIGURE 5. RMSE versus SNR for different µj and σj , To = 30 s andfs = 10Hz.

in Fig. 5, accurate V2I range information is critical for theproposed algorithm. Larger uncertainty level of the estimatedrange information leads to larger RMSEs. Compared withLLs, better performance is achieved for WLLS and with aslightly higher computational complexity.

VI. CONCLUSIONWe propose a low-cost, single on-board receiver and singleRSU-based vehicle localization algorithm. After formulatingthe required V2I information into an over-determined sys-tem, vehicle positions are estimated efficiently by LLS typemethods in a closed-form manner. WLLS-based localizationalgorithm outperforms LLS-based in terms of RMSE andmeter level accuracy is achievable. Furthermore, RMSE per-formance of WLLS is consistent with (42) and approach theCRLB in larger SNR region. Validating the performance ofthe proposed technique via experimental data is one of ourfuture works.

APPENDIX APROOF OF LEMMA 1

Proof: Let χ2n denote the Chi-squared distribution withn degrees of freedom. ej ∼ N (0, σ 2j ), j = 1, 2, · · · , k , is anormal random variable with zeromean and variance σ 2j , thene2j ∼ σ

2j χ

21.

Since themean and variance of χ21 are 1 and 2, respectively.Then we have E[e2j ] = σ

2j and Var[e

2j ] = 2σ

4j . The mean

of b̃j is

E[b̃j] = bj + σ 2j . (56)

Equation (33) is obtained by reformulating (56) into matrixform. Since all odd-order moments of zero-mean Gaussianvariables are zero, then E(ej) = E

(e3j)= 0. The variance of

b̃j is defined as

Var[b̃j] = E[(b̃j − E[b̃j]

)2]= E

[(e2j + 2djej − σ

2j

)2]= E

(e4j)+ 4djE(e3j )+ 2(2d

2j − σ

2j )E(e

2j )

+4σ 2j djE(ej)+ σ4j

= E(e4j)+ 2(2d2j − σ

2j )E(e

2j )+ σ

4j . (57)

Substituting

E(e4j)= Var[e2j ]+

[E(e2j)]2= 3σ 4j , (58)

and E[e2j ] = σ2j into (57), we have

Var[b̃j] = 2σ 4j + 4d2j σ

2j . (59)

Now we compute the covariance of random variables b̃iand b̃j,

Cov[b̃i, b̃j] = E[(b̃i − E[b̃i]

)(b̃j − E[b̃j]

)]= E

[(r2i + 2diri − σ

2i

) (r2j + 2djrj − σ

2j

)]= E

[r2i r

2j

]− σ 2i E

[r2j]− σ 2j E

[r2i]+ σ 2i σ

2j

+ 2djE[r2i rj

]+ 2diE

[rir2j

]+ 4didjE

[rirj]

− 2diσ 2j E [ri]− 2djσ2i E[rj]

= E[r2i r

2j

]− σ 2i σ

2j . (60)

Based on the Isserlis’s theorem [24], we have

E[r2i r

2j

]= E[r2i ]E[r

2j ]+ 2(E[rirj])

2= σ 2i σ

2j . (61)

Because ri and rj are zero mean and independent,then E[rirj] = 0. Substituting (61) into (60) yieldsCov(b̃i, b̃j) = 0, for i 6= j. Diagonal covariance matrix Cb̃in (34) is obtained. �

ACKNOWLEDGMENT(Sugang Ma and Fuxi Wen are co-first authors.)

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SUGANG MA is currently pursuing the Ph.D.degree in computer science with Chang’an Uni-versity. He is currently a Senior Engineer withthe Xi’an University of Posts and Telecommunica-tions. His current research interests include com-puter vision and machine learning. He is a SeniorMember of the China Communications Society.

FUXI WEN received the Ph.D. degree in elec-trical and electronic engineering from NanyangTechnological University, Singapore, in 2013.He held several research positions in Singaporeand U.K., from 2013 to 2017. He has been a MarieSkodowska-Curie Fellow with the Department ofElectrical Engineering, Chalmers University ofTechnology, Sweden, since 2017. His researchinterests include signal processing, 5G localiza-tion, vehicle-to-everything, and their applications

in intelligent and connected vehicles.He is a member of the IEEE Signal Processing Society, the IEEEVehicular

Technology Society, and the International Society of Information Fusion.He currently serves as the Secretary for the IEEE Signal Processing Soci-ety Sweden chapter, a Guest Editor for a Special Issue on Digital SignalProcessing.

XIANGMO ZHAO was with Chang’an Universityfor more than 20 years, where he is currently theVice President and the Director of the Scienceand Technology Innovation Team ofMulti-sourcesTraffic Information Sensing and Fusion, Ministryof Education, and also a Professor with the Schoolof Information Engineering. He has publishedmore than 200 peer-reviewed papers. His researchinterests include the Internet of Vehicles, testingof intelligent vehicles, intelligent transportation

systems, and nondestructive testing for road infrastructures. He currentlyserves as a member of the State Council’s Discipline Evaluation Committeeon Transportation Engineering, and is sitting as the academic leader ofstate-level key discipline of the Traffic Information Engineering and Controlin Chang’an University. He received the National SciTech Progress Awardstwice for his contribution on promoting the development of indoor vehicletesting technology in China. He also received the National May 1st LaborMedal of China, in 2001.

ZHONG-MIN WANG received theM.S. degree inmechatronic engineering and the Ph.D. degree inmechanical manufacture and automation from theBeijing Institute of Technology, Beijing, China,in 1993 and 2000, respectively. He was withXidian University, from 1993 to 1997. From2004 to 2005, he was a Visiting Scholarshipwith the Robotics Laboratory, The AustralianNational University, Canberra, Australia. Since2000, he has been a Full Professor with the School

of Computer Science and Technology, Xi’an University of Posts andTelecommunications.

His current research interests include embedded intelligent perception,intelligent information processing, machine learning, brain–computer inter-face, and effective computing.

DIANGE YANG received the B.S. and Ph.D.degrees from Tsinghua University, in 1996 and2001, respectively, where he is currently a Pro-fessor with the Department of Automotive Engi-neering. He is also the Head of the Departmentof Automotive Engineering. His current researchinterests include intelligent connected vehicles andautonomous driving.

He has authored 12 software copyrights andregistered more than 60 national patents, he also

published 120 papers. He has received numerous awards during his career,including the Excellent Young Scientist of Beijing, in 2010 and the Distin-guished Young Science Technology Talent of Chinese Automobile Industry,in 2011. He is also a recipient of the Second Prize of National TechnologyInvention Rewards of China, in 2010 and 2013.

VOLUME 7, 2019 46121

INTRODUCTIONPROBLEM FORMULATIONPROPOSED LOCALIZATION ALGORITHMLINEAR LEAST SQUARES (LLS)WEIGHTED LLS

PERFORMANCE ANALYSIS BIAS AND MEAN SQUARE ERROR ANALYSISCRAMÉR-RAO LOWER BOUND (CRLB)

SIMULATION RESULTSACCURATE V2I RANGE INFORMATIONINACCURATE V2I RANGE INFORMATION

CONCLUSIONREFERENCESBiographiesSUGANG MAFUXI WENXIANGMO ZHAOZHONG-MIN WANGDIANGE YANG

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