Proactive Doppler Shift Compensation in Vehicular Cyber-Physical Systems
11 Proactive Doppler Shift Compensation inVehicular Cyber-Physical Systems
Jian Du, Xue Liu and Lei Rao
Abstract
In vehicular cyber-physical systems (CPS), safety information, including vehicular speed and loca-tion information, is shared among vehicles via wireless waves at specific frequency. This helps controlvehicle to alleviate traffic congestion and road accidents. However, Doppler shift existing betweenvehicles with high relative speed causes an apparent frequency shift for the received wireless wave,which consequently decreases the reliability of the recovered safety information and jeopardizes thesafety of vehicular CPS. Passive confrontation of Doppler shift at the receiver side is not applicable due tomultiple Doppler shifts at each receiver. In this paper, we provide a proactive Doppler shift compensationalgorithm based on the probabilistic graphical model. Each vehicle pre-compensates its carrier frequencyindividually so that there is no frequency shift from the desired carrier frequency between each pairof transceiver. The pre-compensated offset for each vehicle is computed in a distributed fashion inorder to be adaptive to the distributed and dynamic topology of vehicular CPS. Besides, the updatingprocedure is designed in a broadcasting fashion to reduce communication burden. It is rigorously provedthat the proposed algorithm is convergence guaranteed even for systems with packet drops and randomcommunication delays. Simulations based on real map and transportation data verify the accuracy andconvergence property of the proposed algorithm. It is shown that this method achieves almost the optimalfrequency compensation accuracy with an error approaching the Cram´er-Rao lower bound.
I. I
NTRODUCTION
A. Context and Motivation
Developing vehicles from a purely physical system based on the laws of mechanics andchemistry, to a more sophisticated and intelligent cyber physical system (CPS) with functions
Jian Du and Xue Liu are with the School of Computer Science, McGill University (e-mail: [email protected],[email protected]).Lei Rao is with General Motors, United States (e-mail: [email protected]).
October 3, 2017 DRAFT a r X i v : . [ ee ss . SP ] S e p of communication and control is a promising direction to enhance traffic safety and efficiency.In vehicular CPS, vehicle safety information, e.g., speed, location, and acceleration, are sharedwith high reliability among different vehicles, so that cooperative vehicle control [1] can beapplied to improve the driving safety and alleviate the traffic congestion. The U.S. Departmentof Transportation estimates that vehicular CPS could help address up to percent of crashscenarios with unimpaired drivers, preventing tens of thousands of automobile crashes everyyear [2].Doppler shift, which is the perceived change in frequency of wave emitted by a sourcewhich is moving relative to an observer, exists among vehicles due to their mobility. Sincesafety information is shared via wireless waves at specific frequency, the received waves wouldbe moved from the desired frequency due to Doppler shift, which consequently decreases thereliability of the recovered safety information and thus jeopardizes the safety of vehicular CPS.More specifically, safety information is shared via dedicated short range communications(DSRC) [3] and IEEE 802.11p protocal, which utilize orthogonal frequency division multi-plexing (OFDM) carrier waves to improve spectrum efficiency on 5.9GHz band. AlthoughIEEE 802.11p is considered the de facto standard for on-the-road communications [4], [5],researchers, manufacturers and stakeholder indeed have started to investigate the usability ofLong Term Evolution (LTE), in which orthogonal frequency-division multiple access (OFDMA)is adopted for multiplexing, to support vehicular communications. Interesting readers please refer[5]–[10] and the references therein. Another interesting point is that several auto manufacturersare considering solutions for communication in inter-vehicle communication environments. As Fig. 1. Orthogonal sub-carriers are utilized for vehicle safety information sharing. At the frequency that one sub-carrier takesits peak value, all other sub-carriers are zero. Hence, sampling at frequencies that take peak values is important for safetyinformation recovering at the receiver.
October 3, 2017 DRAFT reported in [5], several original equipment manufacturers have announced agreements withcellular carriers to use equipment from those specific carriers in their vehicles for Internetaccess and other services. This entails the use of a LTE modem installed in cars and theuse of LTE (or LTE-advanced) networks of carriers for several services. Moreover, recently,the Qualcomm Snapdragon automotive development platform, which supports not only IEEE802.11p but also LTE for dedicated short range communications (DSRC) [3], was released [9] toenable auto manufactures, suppliers and developers to rapidly innovate, test and deploy vehicularapplications. The OFDMA signal can be described as a set of closely spaced frequency divisionsub-carriers. In the frequency domain, each sub-carrier is in sinc function form and sub-carriersare allocated to different users. To deliver the safety information, each sub-carrier is modulatedwith a conventional digital modulation scheme (such as QPSK, 16-QAM, etc.) and will berecovered at the receiver. As shown in Fig. 1, though the side lobes of different sinc signalsoverlap with each other, at the peak of each sinc signal, all other sinc signals are zero. Thisfact guarantees that there is no inter-carrier interference if the receiver samples exactly at thesepeak locations. The peak locations may deviate from the pre-defined frequency due to Dopplershift. Because the relative speed between vehicles may be high and results in large Dopplershift, the sampled frequency can not be exactly at the individual peak. Therefore, the sampledvalue contains not only the desired sub-carrier information but also those from other sub-carriersas interferences. Doppler shift would destroy the orthogonal property of different sub-carriers,and it is shown by theoretical analysis and verified by experiments that Doppler shift leads todegradation of system capacity and bit error rate [11].Doppler shift compensation has been studied for one pair transceiver with centralized process-ing method. However, existing solutions [12]–[14] cannot be applied to vehicular CPS due to thefollowing difficulties: 1) For communications between one pair transceiver, the frequency shiftcan be estimated and compensated at the receiver side [12]–[14]. In vehicular CPS, however, ateach receiver safety information from different vehicles arrives at the same time on different sub-carriers, therefore it is necessary to adjust the sampling frequency to compensate frequency offsetscaused by different Doppler frequency shifts. Let i and j denote the transmitter and receiverrespectively. Via training sequence based method [15], each receiver first obtains Doppler shiftestimates f i,j and then adjusts the sampling frequency by multiplying exp( − ι πf i,j tM ) on the t th sample of the received baseband signal [12], with ι denoting the imaginary unit and M denotingthe total parallel subcarriers adopted in the OFDMA scheme. It is evident that when receiver j October 3, 2017 DRAFT receives data from more than one transmitters, it cannot compensate all the Doppler shifts sincethe received signal is a linear superposition of signals from different transmitters. Therefore it isimpossible to adjust the sampling frequency for compensating different frequency shifts. 2) Dueto the moving property of vehicles, the network topology is highly dynamic, and vehicles mayalso randomly join and leave the network. Therefore, a distributed algorithm for frequency shiftcompensation is more suitable than the centralized method to adapt the varying network topology.3) As vehicular CPS may have high density and transmit large volume of data [16], [17], it isprone to resulting in broadcasting storm [18], [19], and thus, an algorithm with communicationoverhead linearly scaling with the vehicle density is desired. To solve the above challenges,distributed algorithm is proposed and is adopted after each receiver obtains the Doppler shiftsestimate with the training based method.
B. Contributions and Organization of the paper
To address above challenges, we propose proactive Doppler shift compensation algorithmbased on the probabilistic graphical model. We assume data are transmitted frame by frame.Each time, when the transmitter sends a data frame out, it is reasonable to assume that Dopplershift for this data frame is a fixed constant due to the fact that the time duration for each dataframe is much smaller than the vehicle speed change. The proposed algorithm compensatesDoppler shift for each data fame in a distributed fashion. We study this algoirthm from bothalgorithm design and theoretical analysis perspectives.From the algorithm design perspective, we construct a probabilistic graphical model to revealthe conditional independence structure of Doppler shifts in vehicular CPS. Though the classicalbelief propagation (BP) algorithm [20] can be applied to distributed frequency shift compensation,the number of messages involved in BP algorithm at each iteration grows quadratically as thenumber of vehicles increases, leading to information network congestion. To overcome thisproblem, we propose a novel distributed algorithm named as linear scaling belief propagation(LSBP) for its linear scalability to network density. We apply LSBP to a vehicular networkwith arbitrary topologies and with potential packet drops as well as random transmission andprocessing delays. It is shown that the total number of messages at each iteration simply equalsto the number of vehicles.From the theoretical analysis perspective, the convergence properties are analyzed for LSBP.Note that though BP has gained great success in many applications, it is found that BP may
October 3, 2017 DRAFT diverge if the network topology contains circles, and the necessary and sufficient convergencecondition is still an open problem. Thus, BP is not reliable for vehicular CPS. In contrast, theanalytical analysis of the proposed LSBP algorithm shows that LSBP is convergence guaranteedfor arbitrary vehicular network topology and is robust to packet drops and random delays.Besides, even with different initial values, the LSBP converges to a unique point. The abovetheoretical analysis is also verified by simulations, and it is shown that the proposed LSBPalgorithm converges quickly with the estimation mean-square-error (MSE) approaching theCram´er-Rao lower bound (CRLB) even under dynamic topologies. Previous works on distributedestimation [21], [22] focus on static network and convergence of standard BP for distributedestimation is analyzed. However, vehicular network is dynamic and may be very dense in certainarea. This paper proposes LSBP algorithm, in which the updating procedure is designed in abroadcasting fashion to reduce communication burden and convergence guaranteed property isanalytically shown.The rest of the paper is organized as follows. A motivating example is shown in Section IIto provide some intuitive insights. The general model and problem formulation are introducedin Section III. The distributed estimation algorism based on probabilistic graphical model ispresented in Section IV. In Section V, convergence property of the proposed algorithm isanalytically proved. Simulation results of proactive frequency shift compensation are illustratedin Section VI. Concluding remarks are given in Section VII.
Notations : Boldface uppercase and lowercase letters represent matrices and vectors, respec-tively. E denotes the statistical expectation operator. A − and A T denote the inverse and thetranspose of matrix A , respectively. Notation N ( x ; µ, P ) stands for the probability densityfunction (PDF) of a Gaussian random variable x with mean µ and variance P . Symbol ∝ represents the linear scalar relationship between two real valued functions, and diag { A } refersto taking the diagonal element of A .II. M OTIVATING E XAMPLE
Combating Doppler shift is also a problem in nature. Certain species of bats, who can produceconstant frequency echolocation calls, compensate for the Doppler shift by lowering their callfrequency as they approach a target. This keeps the returning echo in the same frequencyrange of the normal echolocation call. This dynamic frequency modulation was discovered byHans Schnitzler in 1989 [23]. Inspired by this example, we propose proactive frequency shift
October 3, 2017 DRAFT compensation at each transmitter to mitigate each pair of Doppler shift in vehicular networks.In the following we give a three-vehicle example to explain the main idea.A vehicular CPS consisting of three vehicles as shown in Fig. 2(a) is considered in thismotivating example. It is assumed that vehicles 1, 2 and 3 are within the communication range ofeach other. Vehicle 3 receives safety information broadcasted by vehicles 1 and 2 simultaneously.Doppler shift between vehicles 1 and 3 is designated by f , , and Doppler shift between vehicles 2and 3 is f , . Due to different relative velocities between vehicles, we have f , (cid:54) = f , . Therefore,if vehicle compensates the frequency shift by f , , there is still a frequency shift mismatchbetween vehicles 2 and 3. However, if each vehicle can proactively compensate by certainfrequency amount before sending safety information, it is possible to mitigate the frequencyoffset. For instance, as shown in Fig. 2(b), let the pre-compensated frequency shift at vehicles 1,2 and 3 be f , f , and f , respectively. Then if f , f , and f satisfy f + f = f , , f + f = f , ,and f + f = f , , there will be no frequency offset for each received signal at any vehicle.How to obtain f i ( f , f , and f in this example) is not an easy problem due to the followingchallenges: • f i,j ( f , , f , and f , in this example) cannot be exactly known since the true relativefrequency shift f i,j can only be approximated via statistical estimate [12] or measurements. • Since the number of vehicles is large, the centralized method, which requires the informationof all f i,j and the network topology of vehicular CPS, is difficult to be implemented. Hence,distributed estimation which only involves local computation at each vehicle is desired. • The distributed method needs additional information exchange between neighbors for iter-ative updating, and the number of messages needed for updating should be linear scalingwith the vehicle density. • The distributed estimation algorithm should also be adaptive to the dynamic topology ofvehicular CPS. Besides, convergence of the algorithm has to be guaranteed.III. P
ROBLEM F ORMULATION AND M ODELING
A. Optimal Performance
The interaction topology of a vehicular CPS is represented by an undirected graph G = ( V , E ) ,where V = { , . . . , N } is the set of vehicles, and E ⊆ V × V is the set of communication links.Although we assume the vertices V to be fixed and indexed in a certain order, the mathematicaltheory that follows does not change if the names of the vertices are rearranged. Vehicles within October 3, 2017 DRAFT (a) (b)Fig. 2. Comparison of passive and proactive Doppler shift compensation methods. (a) Passive method: At vehicle , f , and f , cannot be compensated at the same time. (b) Proactive method: Pre-compensating f i at each vehicle results in no relativeDoppler shift between each pair of communication link. communication range of each other are regarded as neighbors, and neighbors of vehicle i aredenoted by B ( i ) (cid:44) { j ∈ V| ( i, j ) ∈ E } . To model the communication link failures, G is assumedto be a Bernoulli network: at each communication, a network link is active with some probability;network links may have different link probabilities; and links fail or are alive independently ofeach other.Let f i,j be the Doppler shift between i and j , then the pre-compensated frequency shift atvehicle i and at vehicle j , i.e., f i and f j , should satisfy f i + f j = f i,j . In practice, we canonly obtain the measurement or estimate [12], [24] of f i,j , denoted as r i,j , between neighboringvehicles { i, j } ∈ E . Thus, we have r i,j = f i + f j + n i,j , (1)where n i,j is the estimation error. It is known that the maximum likelihood estimates of n i,j is asymptotically Gaussian distributed [12], that is, n i,j ∼ N ( n i,j ; 0 , σ i,j ) . Let ˆ f i denote theestimate of f i . The estimation MSE, defined as E { ( ˆ f i − f i ) } , is used to evaluate the performanceof the estimator with the lower bound of MSE given as the performance benchmark. Define f = [ f , f , . . . , f N ] T and stack (1) with respect to all i and j into a matrix form, we obtain r = (cid:104) a A (cid:105) f f + n , (2)where r is a vector containing r i,j with ascending indices first on i and then on j , and n containing n i,j with the indices i , j ordered in the same way as in r . Then, n ∼ N ( n ; , R ) ,where R is a diagonal matrix with σ i,j as diagonal elements which have the same order as r i,j f is set as the reference frequency which can be arbitrary constant October 3, 2017 DRAFT in r . (cid:104) a A (cid:105) is a matrix containing and to make (2) hold for each ( i, j ) ∈ E , and a is itsfirst column. Note that (2) is a standard linear model, so the Cram´er-Rao lower bound (CRLB)of f , which provides the lower bound of the achievable MSE of any unbiased estimator, can beeasily computed as [25] CRLB ( f ) = diag { (cid:0) A T R − A (cid:1) − } . (3)The maximum likelihood estimator is the best linear unbiased estimator approaching CRLB forthe linear model of (2), and is given by [25] [ ˆ f , . . . , ˆ f N ] T (cid:44) arg max f ,...,f N N ( r − f a ; Af , R )= ( A T R − A ) − A T R − ( r − f a ) . (4)Implementing (4), however, not only requires bringing all r i,j and σ i,j to a central computingunit, but also needs the topology of G to construct r and A . Thus, the maximum likelihoodestimator is not scalable with network size, which causes heavy communication burden bytransmitting data from network border to control unit. Besides, (4) needs to be re-computedfrequently due to the dynamic property of vehicular networks. Therefore, distributed estimation,where each vehicle performs estimation with local information, sounds promising [26]. However,achieving the optimal MSE as in (3) in a distributed fashion without global information ischallenging. Leveraging statistical property of { f i } i ∈V for distributed algorithm design is onepromising direction. We next introduce the probabilistic graphical model to reveal conditionalindependence structure of Doppler shifts in vehicular CPS. B. Primer on Probabilistic Graphical Model
In a probabilistic graphical model, each vertex (node) represents a random variable, and thereare a set of edges joining some pairs of vertices. The graph gives a visual way of understandingthe joint distribution of an entire set of random variables on graph [16], [27]. Fig. 3 showsan example of a graphical model for a vehicular CPS with vehicles. Vertex i in the graphcorresponds to f i that needs to pre-compensate on vehicle i . According to (1), the probabilisticrelationship between f i and f j is captured by N ( r i,j ; f i + f j , σ i,j ) and denoted on the graph by anedge linking these two variables. Hence, the probabilistic graphical model has the same networktopology as the vehicular CPS. In this model, the absence of an edge between two vertices hasa special meaning: the corresponding random variables are conditionally independent given onenode’s neighboring nodes. These neighbors are known as Markov blanket , i.e., f i and f j are October 3, 2017 DRAFT
Fig. 3. An example of a graphical model for a vehicular CPS with vehicles. Vertex i in the graph corresponds to f i thatneeds to be pre-compensated on vehicle i . The Markov blanket of node consists of the set of its neighbouring nodes { , , } .Besides, messages from to its different neighbors are different in BP algorithm, and these messages are denoted by differentcolors. conditional independent given all { f k } k ∈B ( i ) or all { f k } k ∈B ( j ) . For example, as shown in Fig.3, f and f are conditional independent given { f , f , f } . Thus, it is possible to obtain theestimate of f with the help of its neighbors, i.e., { , , } , via message exchange.IV. D ISTRIBUTED A LGORITHM D ESIGN
In virtue of the conditional independence relationship between variables as revealed by theprobabilistic graphical model, distributed inference can be designed with only local informa-tion between neighbors. In this section, leveraging the probabilistic graphical model, beliefpropagation (BP) algorithm is studied first for estimation of pre-compensated frequency shift.Inspired by BP, a distributed estimation algorithm named as linear scaling BP (LSBP), whichhas low communication overhead and is convergence guaranteed, is then proposed. Notice thatcommunication scheme [28] that is robust to Doppler shift can be adopted for message exchangebefore Doppler shifts are compensated.
A. Belief Propagation Algorithm
With Gaussian belief propagation (BP) [20] algorithm for linear Gaussian model, at everyiteration, each node sends a (different) message to each of its neighbors and receives a messagefrom each neighbor. The message from vehicle j to vehicle i is defined as the product of the October 3, 2017 DRAFT0 local function N ( f i,j ; f i + f j , σ i,j ) with messages received from all neighbors except i , and thenmaximized over all involved variables except f i . Mathematically, it is defined as m ( l ) j → i ( f i ) = max f j N ( f i,j ; f i + f j , σ i,j ) × (cid:89) k ∈B ( j ) \ i m ( l − k → j ( f j ) . (5)The message m ( l ) j → i ( f i ) is computed and exchanged among neighbors. One possible schedulingfor message exchange is that all vehicles perform local computation and message exchange inparallel [29]. In any round of message exchange, a belief of f i can be computed at each vehicle i locally, as the product of all the incoming messages from neighbors, which is given by b ( l ) ( f i ) = (cid:89) j ∈B ( i ) m ( l ) j → i ( f i ) . (6)The belief b ( l ) ( f i ) serves as the approximation of max { f ,...,f N }\ f i N ( r ; Af , R ) . Therefore, theestimate of f i in the l th iteration can be computed by ˆ f ( l ) i = max f i b ( l ) ( f i ) . (7)Note that the message and belief updating rules denoted by (5) and (6) are naturally distributed:message m ( l ) j → i ( f i ) in (5) is computed internally by j , and then sent to its neighbor i . Afterreceiving all the messages from its neighbors, i computes the belief b ( l ) ( f i ) according to (6).The message exchange between neighboring vehicles can be realized via local communica-tions; however, as packet drops and random delays are the bottleneck of communication invehicular CPS [30], their impact on message exchange should be addressed. To do so, totallyasynchronous scheduling is adopted. More specifically, each vehicle still performs message andbelief computations at the individual predefined time, even when it doesn’t receive newly updatedmessages from some of its neighbors. This totally asynchronous scheduling is defined as follows. Definition 1 (Totally Asynchronous Scheduling): The message available to j at time l is m ( τ k → j ( l )) k → j with k ∈ B ( j ) , where τ k → j ( l ) satisfies ≤ τ k → j ( l ) ≤ l , and lim l → + ∞ τ k → j ( l ) = + ∞ for all { k, j } ∈ E . The physical meaning of the above definition is that, even though packet drops and randomdelays may cause some updated messages failed to be received, local computation at each vehiclecan still continue with part of the updated messages and part of the outdated messages receivedat the last iteration. The outdated messages can eventually be replaced by successfully received
October 3, 2017 DRAFT1 messages in the future updating. Each vehicle j keeps a buffer with the most recently receivedmessages from all its neighbors, i.e., m ( τ k → j ( l − k → j ( f j ) at iteration time l . Therefore, under packetdrops and random delays, the outgoing message m ( l ) j → i ( f i ) in (5) can be computed as m ( l ) j → i ( f i ) = max f j N ( f i,j ; f i + f j , σ i,j ) × (cid:89) k ∈B ( j ) \ i m ( τ k → j ( l − k → j ( f j ) . (8)Similarly, the belief in (6) can be computed as b ( l ) ( f i ) = (cid:89) j ∈B ( i ) m ( τ k → j ( l )) j → i ( f i ) . (9) B. Message Computation for BP
From (8) and (9), m ( l ) j → i ( f i ) and b ( l ) ( f i ) are functions of variable f i , and they represent theestimate of f i by j and i , respectively. As these messages are updated at each iteration, explicitexpressions of these messages are needed. First, to facilitate the subsequent updating, the initialmessage is set to be in Gaussian function form i.e., N ( f i ; η (0) j → i , C (0) j → i ) .Next, m ( l ) j → i ( f i ) is computed. Since N ( f i,j ; f i + f j , σ i,j ) is a Gaussian function, according to(5), m (1) j → i ( f i ) is also a Gaussian function, and by induction, it can be easily proved that m ( l ) j → i ( f i ) in (8) keeps Gaussian form for arbitrary l . Therefore, only its mean and variance need to betransmitted for exchanging the message m ( l ) j → i ( f i ) .At this point, we can compute the messages at any iteration. In general, for the l th ( l =2 , , · · · ) round of message exchange, let the available messages variance and mean at from k to j are (cid:2) C ( τ k → j ( l − k → j (cid:3) − and η ( τ k → j ( l − k → j , respectivly. Then, j computes and transmits theoutgoing messages to each of its neighbors individually. After some straightforward but tediousderivations, reciprocal of the message variance is given by (cid:2) C ( l ) j → i (cid:3) − = (cid:2) σ i,j + (cid:2) (cid:88) k ∈B ( j ) \ i (cid:2) C ( τ k → j ( l − k → j (cid:3) − (cid:3) − (cid:3) − , (10)and the message mean is expressed as η ( l ) j → i = (cid:110) r i,j + (cid:2) (cid:88) k ∈B ( j ) \ i (cid:2) C ( τ k → j ( l − k → j (cid:3) − (cid:3) − (11) × (cid:2) (cid:88) k ∈B ( j ) \ i (cid:2) C ( τ k → j ( l − k → j (cid:3) − η ( τ k → j ( l − k → j (cid:3)(cid:111) . October 3, 2017 DRAFT2
Due to packet drops and random delays, (cid:2) C ( l ) j → i (cid:3) − and η ( l ) j → i , may or may not be successfullyreceived by i for updating b ( l ) ( f i ) . Following Definition 1, (cid:2) C ( τ j → i ( l )) j → i (cid:3) − and η ( τ k → j ( l − j → i are usedto denote the available information of i at iteration l , then i can compute the BP estimates via(9), which can be easily shown to be b ( l ) i ( f i ) ∝ N ( f i | µ ( l ) i , P ( l ) i ) , with (cid:2) P ( l ) i (cid:3) − = (cid:88) j ∈B ( i ) (cid:2) C ( τ j → i ( l )) j → i (cid:3) − , (12)and ˆ f ( l ) i (cid:44) µ ( l ) i = P ( l ) i (cid:88) j ∈B ( i ) (cid:2) C ( τ k → j ( l − j → i (cid:3) − η ( τ k → j ( l − j → i . (13)Let l max denote the maximum updating times for each vehicle, and the algorithm terminateswhen the maximum number of iteration l max is reached, or when ∆ i (cid:44) (cid:107) ˆ v ( l ) i − ˆ v ( l − i (cid:107) < th , where th is a threshold. The BP algorithm for proactive Doppler shift compensation is summarized inAlgorithm 1.It is well known that the convergence of BP is not guaranteed for topology with loops.Consequently, the BP algorithm may either converge or diverge, resulting in unreliable estimates.Moreover, it is apparent that the outgoing messages, i.e., (10) and (11), to different neighborsare different, and thus, huge amount of information is broadcasted in the network. Such problemis especially serious in dense traffic and leads to information network traffic congestion [18],[19].To address the above problems, in the next section, we design a novel distributed algorithm,which not only guarantees the iterative updating convergence but also has the property that theamount of information exchange among vehicles is linear to the traffic density. C. Design of Linear Scaling BP
To get some insights on low communication overhead message passing algorithm, we startby investigating BP in the simplest possible graph: a tree graphical model. In this model, BPcomputes the maximum likelihood estimate in an efficient way with convergence guaranteed.In a tree graphical model as shown in Fig 4, for each pair of variables connected by an edge,the variable near the root is named as parent , while the other variable is named as child . Then,the messages of BP can be categorized into two kinds: one is from parent to child denotedas m ( l ) p → c ( f c ) , and the other is from child to parent denoted as m ( l ) c → p ( f p ) . Then, we have thefollowing property. October 3, 2017 DRAFT3
Fig. 4. An example of probabilistic graphical model with tree topology.
Property 1
For a tree topology vehicular network with root being the reference vehicle, the BPupdating equation (5) equals m ( l ) p → c ( f c ) = max f p N ( f p,c ; f p + f c , σ p,c ) b ( l − p ( f p ) for message fromparent to child, and m ( l ) c → p ( f c ) is a constant for message from child to parent. Proof 1
See Appendix A.
With Property 1, we can exactly compute the maximum likelihood function for the tree networktopology. Then, we apply it to a network containing loops. By simply generalizing Property 1,the message in a loopy graph is computed by ˜ m ( l ) j → i ( f i ) = max f j N ( f i,j ; f i + f j , σ i,j ) b ( l − j ( f j ) ,and the outgoing message is ˜ b ( l ) ( f i ) = (cid:81) j ∈B ( i ) ˜ m ( l ) j → i ( f i ) .For networks with random delays and packet drops, the message updating equation can beeasily obtained as ˜ m ( l ) j → i ( f i ) = max f j N ( f i,j ; f i + f j , σ i,j ) b ( τ j → i ( l − j ( f j ) , (14)and the outgoing message is ˜ b ( l ) ( f i ) = (cid:89) j ∈B ( i ) ˜ m ( l ) j → i ( f i ) . (15)Note that (14) differs from the standard BP of (8) in that each vehicle only transmits ˜ b ( l − ( f j ) to all its neighbors at one time, then ˜ m ( l ) j → i ( f i ) is computed at node i and then the belief ˜ b ( l ) ( f i ) can be obtained according to (14). Because the message need to be transmitted at each iterationequals the number of vehicles, the proposed method is named as linear scaling BP (LSBP).Fig. 5 denotes the message passing with LSBP. Notice that, the message computation equations((15) and (16)) for each node only depend on message transmitted from neighbor nodes and areindependent of network topology, which further implies that there is no need to construct a treetopology for message scheduling. Next, the explicit message expression of LSBP is computed. October 3, 2017 DRAFT4
Fig. 5. The same example of a graphical model for a vehicular CPS as shown in Fig. 3. However, the outgoing messages from to different neighbors are the same in LSBP algorithm. D. Message Computation for Linear Scaling BP
To start the recursion, in the first round of message exchange, the initial incoming message issettled as b ( τ j → i ( l − j ( f j ) = N ( f j ; µ (0) j , P (0) j ) , with P (0) j > and µ (0) j can be arbitrary value. Since N ( f i,j ; f i + f j , σ i,j ) is a Gaussian pdf, according to (14), ˜ m (1) j → i ( f i ) is still a Gaussian function.In addition, ˜ b (1) ( f i ) , being the product of Gaussian functions in (15), is also a Gaussian function[31], [38]. Consequently, in LSBP, during each round of message exchange, all the messagesare Gaussian functions, and only the mean and the variance need to be exchanged betweenneighbors.At this point, we can compute the messages of LSBP at any iteration. In general, in the l th ( l = 2 , , · · · ) round of message exchange, vehicle i with the available message b ( τ j → i ( l − j ( f j ) ∝N ( f j ; µ ( τ j → i ( l − j , P ( τ j → i ( l − j ) from its neighbors, computes the outgoing messages via (14). Byputting the explicit expression of b τ j → i ( l − j ( f j ) into (14) and after some tedious but straightfor-ward computations, we have ˜ m ( l ) j → i ( f i ) ∝ N ( f i ; η ( l ) j → i , C ( l ) j → i ) in which C ( l ) j → i = σ i,j + P ( τ j → i ( l − j , (16)and η ( l ) j → i = f i,j + µ ( τ j → i ( l − j . (17)Furthermore, during each round of message exchange, each vehicle computes the belief for f i via (15), which can be easily shown to be ˜ b ( l ) i ( f i ) ∝ N ( f i ; µ ( l ) i , P ( l ) i ) , with variance P ( l ) i = (cid:2) (cid:88) j ∈B ( i ) (cid:2) C ( l ) j → i (cid:3) − (cid:3) − , (18) October 3, 2017 DRAFT5 and mean µ ( l ) i = P ( l ) i (cid:8) (cid:88) j ∈B ( i ) (cid:2) C ( l ) j → i (cid:3) − η ( l ) j → i (cid:9) . (19)The updating is iterated between (16), (17) and (18), (19) at each vehicle in parallel. One wayto terminate the iterative algorithm is that all vehicles stop updating when a predefined maximumnumber of iterations l max is reached. Since LSBP is convergence guaranteed as proved in thenext section, the termination can also be implemented once the algorithm converged. The LSBPalgorithm is summarized in Algorithm 2.Table I shows messages need to be computed and transmitted at each vehicle at each iterationfor BP and LSBP. It can be easily concluded that in contrast to BP algorithm, with which theamount of messages need to be computed and transmitted by each vehicle at each iteration isproportional to the number of neighbors, with LSBP each vehicle only needs to compute andtransmit one pair of mean and variance to all its neighbors. Therefore, LSBP is scalable withtraffic density. Moreover, in a limit case where G is a fully connected graph, i.e., |B ( i ) | = N − ,as can be seen from Table I, the number of messages exchanged in the network with BP is ( N − N . Thus the total number of messages, grows quadratically when the vehicle number N increases, leading to information network congestion. While with LSBP it is only N . Therefore,the number of messages involved in BP increases much faster than that with LSBP which leavesthe network vulnerable to information congestion. To get further insights of the proposed LSBPalgorithm, its convergence property is studied in the following section.V. C ONVERGENCE A NALYSIS FOR
LSBPAs BP may diverge if the network topology contains circles [32], [33], which is often the casein vehicular CPS, BP is not reliable. In this section, we analytically proved that the proposedLSBP algorithm is convergence guaranteed with feasible initial values, and µ ( l ) j and P ( l ) j convergeto the same fixed point respectively even with different initial value pairs µ (0) j and P (0) j . Due to TABLE IC
OMPLEXITY P ER E STIMATION U PDATE
BP LSBP (cid:2) C ( τ i → j ( l )) i → j (cid:3) − , η ( τ i → j ( l )) i → j , ∀ j ∈ B ( i ) P ( l ) i , µ ( l ) i October 3, 2017 DRAFT6 the estimate by LSBP shown in (19) depends on P ( l ) i and η ( l ) j → i , we first prove the convergenceof P ( l ) i and then η ( l ) j → i . A. Convergence of Message Variance
By substituting (16) into (18), the updating equation of P ( l ) i is given by (cid:2) P ( l ) i (cid:3) − = (cid:88) j ∈B ( i ) (cid:2) σ i,j + P ( τ j → i ( l − j (cid:3) − . (20)Let p ( l ) be a vector containing of all the message variance at the l th iteration, i.e., p ( l ) (cid:44) [[ P ( l )2 ] − , [ P ( l )3 ] − , . . . , [ P ( l ) N ] − ] T and define an evolution function F as p ( l +1) = F ( p ( l ) ) . We willsay that a p (0) > is a feasible initial value if p (0) > satisfies F ( p (0) ) ≥ p (0) ) or F ( p (0) ) ≤ p (0) .Notice that one easy obtained feasible p (0) is by setting (cid:2) P ( l ) i (cid:3) − = 0 . Next, it is shown that thefunction F ( · ) has the following properties for arbitrary p (0) > . Property 2
The following claims hold with l ∈ { , , · · · } :P2-1. Positive limited range: F ( ) > F ( p ( l ) ) > .P2-2. Scalability: ∀ α > , α F ( p ( l ) ) > F ( α p ( l ) ) .P2-3. Monotonicity: if p ( l ) ≥ ˜ p ( l ) then F ( p ( l ) ) ≥ F ( ˜ p ( l ) ) . Proof 2
See Appendix B
Then we can prove the convergence property of the belief variance in LSBP.
Theorem 1
With arbitrary feasible initial value P (0) i , the belief variance P ( l ) i of LSBP shownin (20) converges to a unique fixed point for a specific network topology. Proof 3
For arbitrary feasible initial variance P (0) i after the first round updating, we have p (1) ≥ p (0) or p (1) ≤ p (0) . We first investigate the case when p (1) ≥ p (0) . According to P2-3,we have F ( p (1) ) > F ( p (0) ) or equivalently p (2) > p (1) . Then, the monotonic increasing propertyof p ( l ) can be proved by induction following P2-3. According to P2-1, p ( l ) is upper bounded by F ( ) . From the monotone convergence theorem [ ? ], therefore, p ( l ) is convergence guaranteed.With the same argument, we can prove that if p (1) ≤ p (0) , p ( l ) is a monotone decreasing positivesequence, which is convergence guaranteed.In the subsequent, the unique property of the converged p ( l ) for a specify network topology isproved by contradiction. Suppose p ∗ and ˜ p ∗ are two distinctive fixed point, and without loss of October 3, 2017 DRAFT7 generality assume p ∗ > ˜ p ∗ . Due to the elements in p and ˜ p are all positive, there exists α > such that α ˜ p ∗ ≥ p ∗ and for some particular index i that α ˜ P ∗ i = P ∗ i . (21) On the other side, following the definition of fixed point, we have p ∗ ( i ) = F ( p ∗ )( i ) ≤ F ( α ˜ p ∗ )( i ) ,where the inequality comes from the monotonic property (P2-3). Then following the scalabilityproperty (P2-2), we have P ∗ j < α ˜ P ∗ j . (22) Hence, (21) and (22) is a contradiction, and p ∗ and ˜ p ∗ are the same fixed point a specify networktopology. Therefore the element P ( l ) i in p ( l ) converge to a fixed positive value. This completesthe proof. Next, we focus on the convergence property of the estimate µ ( l ) i with the conclusion that P ( l ) i has converged. B. Convergence of Message Mean
Suppose the converged value of P ( l ) j is P ∗ j , then following (16), we have C ( l ) j → i = σ i,j + P ∗ j .Thus, C ( l ) j → i is also convergence guaranteed, and then the converged value is denoted by C ∗ j → i .Putting C ∗ j → i into (18) and substituting the result into (19), we have µ ( l ) i = [ (cid:88) j ∈B ( i ) (cid:2) C ∗ j → i (cid:3) − ] − (cid:110) (cid:88) j ∈B ( i ) (cid:2) C ∗ j → i (cid:3) − ( r ij − µ ( l − j ) (cid:111) . (23)In the subsequent, we prove the following theorem for the convergence property of µ ( l ) i . Theorem 2
For asynchronous updating, with feasible initial P (0) j , the mean of LSBP algorithm,i.e., µ ( l ) i in (23), converges to a fixed point irrespective of the network topology. Proof 4
Let K ji (cid:44) [ (cid:80) j ∈B ( i ) (cid:2) C ∗ j → i (cid:3) − ] − (cid:2) C ∗ j → i (cid:3) − , and ξ i (cid:44) [ (cid:80) j ∈B ( i ) (cid:2) C ∗ j → i (cid:3) − ] − (cid:110) (cid:80) j ∈B ( i ) (cid:2) C ∗ j → i (cid:3) − r ij (cid:111) ,then (23) can be expressed as µ ( l ) i = ξ i − (cid:88) j ∈B ( i ) K j,i µ ( l − j . (24) Due to the fact that f is the reference for pre-compensated frequency shift estimation, thus µ ( l )1 is a constant which is denoted by µ , and then only the convergence of µ ( l )2 , µ ( l )3 , . . . , µ ( l ) N needs October 3, 2017 DRAFT8 to be investigated. Hence, we separate µ from (cid:80) j ∈B ( i ) , K j,i µ ( l − j in (24), and the result can beexpressed as µ ( l ) i = ( ξ i − K ,i µ ,i ) − (cid:88) j ∈{B ( i ) \ } j,i K j,i µ ( l − j , (25) where j,i is an indicator random variable with j,i = 1 if { j, i } ∈ E otherwise is j,i = 0 .Next, the convergence of µ ( l )2 , µ ( l )3 , . . . , µ ( l ) N will be investigated all together. Define µ ( l ) =[ µ ( l )2 , µ ( l )3 , . . . , µ ( l ) N ] T , and k i = [ ,i K ,i , ,i K ,i , . . . , N,i K N,i ] T , and then (27) can be reformu-lated as µ ( l ) i = ( ξ i − K ,i µ ,i ) − k Ti µ ( l − . (26) Piling up (26) for all µ i with the increasing order on i , we obtain the updating equation forall µ as µ ( l ) = η − Kµ ( l − , (27) where η = [ ξ − K , µ , , ξ − K , µ , , . . . ] T and K is an ( N − × ( N − matrixwith the i th row of K being k Ti . According to the definition of k i above (26), the summation of k i can be written as (cid:80) j ∈B ( i ) \ K j,i = (cid:80) j ∈B ( i ) \ (cid:2) C ∗ j → i (cid:3) − / (cid:80) j ∈B ( i ) (cid:2) C ∗ j → i (cid:3) − . It is obvious that,if ∈ B ( i ) , (cid:80) j ∈B ( i ) \ K j,i < , and if (cid:54)∈ B ( i ) , (cid:80) j ∈B ( i ) \ K j,i ≤ . Therefore, K is a non-negative matrix having row sums less than or equal to with at least one row sum less than .Hence, K is a substochastic matrix. Consequently, K in (27) is a non-negative and irreduciblesubstochastic matrix, therefore, ρ ( | K | ) = ρ ( K ) < , where ρ ( · ) denotes the spectrum radius ofa matrix. Then (27) is convergence guaranteed [34]. Hence, the convergence of µ ( l ) i in (23) isguaranteed irrespective the network topology. VI. E
XPERIMENT E VALUATIONS
In this section, realistic data traces and simulation tools are employed to evaluate the proposedalgorithm for proactive Doppler shift compensation. As shown in Fig. 6, a real street mapcovering a km × km area of Montreal is generated from OpenStreetMap [35]. Hereafter, andunless stated otherwise, vehicles are generated on the map by simulation tool SUMO [36].The traffic data generated by SUMO includes vehicular positions, destinations, travelling pathsand speeds. These parameters are also within practical limitations as in Fig. 6. For example,vehicle speeds are within the speed limitation of corresponding street. According to [3], thecommunication range of each vehicle is set to be m. The true Doppler shift between each October 3, 2017 DRAFT9
Fig. 6. The street map covering a km × km area of Montreal from OpenStreetMap [35], with vehicles generated bySUMO [36]. The yellow triangles stand for vehicles running on streets. pair of vehicles within communication range can be computed according to f i + f j = v i,j f /c ,where v i,j is the relative velocity between vehicles i and j , f is the carrier frequency, and c isthe speed of waves.In practice, message exchange between vehicles may fail due to various factors, like sep-aration distance, signal propagation environment, received signal strength, transmission powerand modulation rate [30]. In the following experiments, different packet delivery ratio (PDR),which is the ratio of the number of packets successfully delivered to destination compared tothe number of packets that have been sent out by the transmitter, is set to show the impacts ofpacket drop on proposed algorithms.First, the convergence property of P ( l ) i as shown in Theorem 1 is verified by simulations.The network topology is randomly generated by SUMO, and PDR is set to be . The initialmessage variance for each P (0) i is set to be , , , . and . , respectively. The convergenceproperty of P ( l )6 is demonstrated in Fig. 7 as an example. It is clear that though P ( l )6 keepsmonotonic increasing or decreasing with different initial values, they converge to the same point.Thus, the conclusion of Theorem 1 is verified by simulations that with arbitrary feasible initialvalue, the belief variance P ( l ) i of LSBP shown in (20) converges to a unique fixed point.Next, the accuracy and convergence property of ˆ f i is investigated. Average MSE, definedas N (cid:80) Ni =1 E { ( ˆ f i − f i B ) } , is adopted as the performance criteria. Fig. 8 shows that for differentPDRs ( and ), the convergence speeds of BP and LSBP algorithms differ. Nevertheless,even for PDR as low as , both BP and LSBP converge to a fixed estimate point within iterations, and thus, they are robust to packet drops. Besides, LSBP has the MSE performancethat approaches the CRLB. Note that BP can also reach the CRLB as shown in Fig. 8, but its October 3, 2017 DRAFT0 −3 −2 −1 Nomuber of iterations V a r i a n ce Initial variance 10 Initial variance 10Initial variance 1Initial variance 10 −1 Initial variance 10 −3 Fig. 7. Convergence property of P ( l )6 for different initial values. −4 −2 Number of iterations M S E o f fr e qu e n c y s h i f t LSBP with PDR 60%BP with PDR 60%BP with PDR 80%LSBP with PDR 80%Lower bound
Fig. 8. Accuracy and convergence property of ˆ f i under different PDRs. convergence for loopy topology network is not guaranteed, and its communication overhead islarge as shown in Table I.Fig. 9 shows adaptiveness property of the proposed algorithms to the dynamic topology ofvehicular networks. At first, the network topology is the same as that adopted in Fig. 8. Atiteration , vehicles , , and leave the network, and at iterations and , new vehiclesjoin the network at former positions of , , and , respectively. It can be seen that theaverage MSE increases at iteration due to vehicles’ leaving, and it decreases after iteration because new vehicles join in and bring new measurements. It is shown that the impact ofvehicles’ leaving and joining on the performance of BP and LSBP is very trivial, and bothalgorithms are adaptive to topology varying.In the following, the communication burden imposed by BP and LSBP are analyzed andcompared. First, the total number of messages transmitted among vehicles at each iteration for October 3, 2017 DRAFT1 −2 Number of iterations M S E o f fr e qu e n c y s h i f t LSBP BP Lower bound
Fig. 9. Adaptive property of proposed algorithms to dynamic vehicular network. At iteration , vehicle , , and leavethe network, and at iterations and , new vehicles join the network at former positions of , , and , respectively.
70 80 90 100 110 1200500100015002000 Vehicle number N u m b e r o f m e ss a g e s BPLSBP
Fig. 10. Comparison on the total number of messages transmitted in a vehicular CPS at each iteration for BP and LSBP.
70 80 90 100 110 120051015 Vehicle number N u m b e r o f it e r a ti on s BPLSBP
Fig. 11. Iteration numbers upon convergence versus the vehicle number.
BP and LSBP is compared in Fig. 10. It is shown that as the vehicle number increases from to , the number of messages required to be transmitted in BP increases quickly, which may leadto information network congestion [18], [37]. It verifies the analysis of Table I that the number October 3, 2017 DRAFT2
Number of iterations -2 -1 M S E o f fr e qu e n c y s h i f t Vehical 1Vehical 2
Fig. 12. Iteration numbers upon convergence versus the vehicle number. of messages increases quadratically for dense network. In contrast, the number of messagesinvolved in LSBP increases mildly, which, in fact, is simply equal to the number of vehicles.Next, the number of iterations with BP and LSBP for different scales of vehicles are studied.As shown in Fig. 11, for both BP and LSBP algorithms, the iteration number upon convergenceincreases mildly with the increasing number of vehicles. This property makes sure that proactivefrequency compensation can be achieved within limited time in vehicular CPS. From both Figs.10 and 11, we can conclude that LSBP has much lower communication overhead compared withBP and is much preferred in dense traffic networks. Furthermore, the LSBP algorithm proposeddoes not need a control center and control channel to coordinate the distributed computing orperform scheduling. Each computation is performed locally and information is only needed to betransmitted to the direct neighbors. The overhead is very small compared with communicationinformation that not only includes safety information but also transmission data for social networkand entertainment such as video/voice data [5]–[10].Next, we show that the overhead of LSBP is reasonable and practical. We adopt double-precision floating-point format which uses bytes to represent decimal fraction. Since each timeonly two real value scalars (mean and variance) need to be transmitted at each node, informationneeds to be transmitted are bytes. According to the empirical measurement in [30], A -Byte packet takes . ms transmission time using Mbps (QPSK) in vehicular networks. It isevident that time needed to transmit the above bytes in each iteration is smaller than . ms.Take a network with vehicles as an example, as shown in Fig. 11, the average number ofiteration is around . Since the proposed method is a parallel algorithm, the transmission time October 3, 2017 DRAFT3 is smaller than . × . ms, which is a acceptable overhead for wireless communication.At last, we show that even for highway environment where the network topology is linearand the nodes are sparsely connected the proposed LSBP algorithm still works. We assume vehicles scattered as a line in a highway and each vehicle’s speed is generated by SUMOsimulations. It is assumed that each vehicle can only communicate with its front and backneighbors. The convergence performance of estimation MSE is given in Fig. 12. It is shown thatit is less accurate than the dense network topology case that we considered in Fig. 8. But theperformance is accurate enough for further data detection.VII. C ONCLUSIONS
In this paper, an algorithm for proactive Doppler shift compensation has been proposed toenhance the reliability of safety information sharing in vehicular cyber-physical systems. Prob-abilistic graphical model has been incorporated to reveal the conditional independence propertyof the pre-compensated frequency offset at each vehicle. In this distributed message passingalgorithm, named as linear scaling belief propagation (LSBP), the communication overhead islinear scaling with the network density. Analytical analysis has been conducted to rigorouslyprove that the the proposed algorithm is convergence guaranteed wit feasible initial values evenfor systems with packet drops and random delays. Though LSBP only requires local informationat each vehicle, simulations based on real map and transportation data have verified that LSBPachieves almost the optimal frequency compensation accuracy with an error approaching theCram´er-Rao lower bound. Simulations also show that the number of exchanged messages linearlyscales with the number of vehicles, and the iteration number upon convergence increases mildly,and thus, implementing LSBP imposes tolerable communication overhead.A
PPENDIX AP ROOF OF P ROPERTY m ( l ) p → c ( f c ) and m ( l ) c → p ( f p ) are independent, sowe compute them separately. October 3, 2017 DRAFT4
First, we compute m ( l ) c → p ( f c ) by starting from a variable c without any child. According to (5),we have m ( l ) c → p ( f p ) = max f c ψ p,c ( f p , f c )= max f c N ( r p,c ; f p + f c , σ p,c )= max f c N ( f c ; r p,c − f p , σ p,c )= σ − p,c / √ π, (28)where the third equation is from N ( x ; µ, σ ) = N ( µ ; x, σ ) , and the fourth equation comes fromthe fact that maximum value of Gaussian PDF only relates to its variance. Next, we computethe message from p to its parent p (cid:48) . According to (5), we have m ( l ) p → p (cid:48) ( f p (cid:48) ) = max f p ψ p (cid:48) ,p ( f p (cid:48) , f p ) (cid:89) c ∈B ( p ) \ p (cid:48) m ( l − c → p ( f p ) ∝ max f p N ( f p ; f p (cid:48) + r p (cid:48) ,p , σ p (cid:48) ,p )= σ − p (cid:48) ,p / √ π, (29)where the first equation is due to m ( l ) c → p ( f p ) is a constant. Then, by induction, we have m ( l ) c → p ( f c ) is constant for all messages from child to parent. Therefore, this kind of message can be omittedfor computation and transmission, and only m ( l ) p → c ( f ( c ) ) needs to be computed.Following (5), we obtain m ( l ) p → c ( f c ) = max f p ψ p,c ( f p , f c ) (cid:89) p (cid:48) ∈B ( p ) \ c m ( l − p (cid:48) → p ( f p )= max f p ψ p,c ( f p , f c ) b ( l − p ( f p ) . (30)This completes the proof. A PPENDIX BP ROOF OF P ROPERTY σ i,j > and P (0) j > , according to (20), it is obvious that [ P (1) i ] − > . Then, it can be easily proved by induction that for arbitrary l , (cid:2) P ( l ) i (cid:3) − > , andthus p ( l +1) = F ( p ( l ) ) > . Then, if p (0) > , by induction we have p ( l ) > for l ∈ { , , , . . . } .Furthermore, according to (20) it is shown that (cid:2) P ( l ) i (cid:3) − is a monotonic decreasing functionwith respect to P ( τ j → i ( l − j . As (cid:2) P ( l ) i (cid:3) − > or equivalently P ( l ) i > , we have (cid:2) P ( l ) i (cid:3) − < (cid:80) j ∈B ( i ) (cid:2) σ i,j + 0 (cid:3) − or equivalently F ( ) > F ( p ( l ) ) > . Hence, P2-1 is proved. October 3, 2017 DRAFT5
Next, we prove P2-2. Let F i ( p ( l ) ) denote the i th element in F ( p ( l ) ) , then according to (20),for arbitrary α > , we have α F i ( p ( l ) ) = α (cid:88) j ∈B ( i ) (cid:2) σ i,j + P ( l ) j (cid:3) − . (31)Besides, the corresponding i th element in F ( α p ) is given by F i ( α p ( l ) ) = (cid:88) j ∈B ( i ) (cid:2) σ i,j + P ( l ) j α (cid:3) − . (32)Computing (31)-(32), we have α F i ( p ( l ) ) − F i ( α p ( l ) )= α (cid:88) j ∈B ( i ) (cid:8)(cid:2) σ i,j + P ( l ) j (cid:3) − − (cid:2) ασ i,j + P ( l ) j (cid:3) − (cid:9) . (33)As α > and σ i,j > , it can be concluded that in (33), α F i ( p ( l ) ) − F i ( α p ( l ) ) > . The aboveinequality is satisfied for arbitrary i , so we have ∀ α > , α F ( p ) > F ( α p ) . Thus, the scalabilityis proved.At last, we prove the monotonic property (P2-3). Denote p ( l ) = [[ P ( l )2 ] − , [ P ( l )3 ] − , . . . , [ P ( l ) N ] − ] and ˜ p ( l ) = [[ ˜ P ( l )2 ] − , [ ˜ P ( l )3 ] − , . . . , [ ˜ P ( l ) N ] − ] . If p ( l ) ≥ ˜ p ( l ) , we have (cid:80) j ∈B ( i ) (cid:2) σ i,j + P ( l ) j (cid:3) − ≥ (cid:80) j ∈B ( i ) (cid:2) σ i,j + ˜ P ( l ) j (cid:3) − .Then, according to (20), (cid:2) P ( l +1) i (cid:3) − ≥ (cid:2) ˜ P ( l +1) i (cid:3) − . Therefore, F ( p ( l ) ) ≥ F ( ˜ p ( l ) ) . The monotonicproperty is proved. R EFERENCES [1] R. M. Murray, “Recent research in cooperative control of multivehicle systems,”
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October 3, 2017 DRAFT8
Algorithm 1
BP for Proactive Doppler Shift Compensation Initialize: Set the initial incoming message parameters (cid:2) C (0) k → j (cid:3) − > and η ( τ k → j (0)) k → j can bearbitrary value for all j ∈ V and { k, j } ∈ E ; for l ∈ { st , nd , . . . , l thmax } iteration do Vehicle j with j = 1 , · · · , N in parallel Compute the outgoing messages (cid:2) C ( l ) j → i (cid:3) − and η ( l ) j → i to all neighbors i ∈ B ( j ) individually,via (10) and (11); Transmit (cid:2) C ( l ) j → i (cid:3) − and η ( l ) j → i to each neighbor i ∈ B ( j ) , separately; With the available (cid:2) C ( τ j → i ( l )) j → i (cid:3) − and η ( τ k → j ( l )) j → i , i computes the estimate ˆ f ( l ) i via (13); end parallel If ∆ i < th , return current estimate ˆ f ( l ) i ; end for If ∆ i > th , BP does not converge. Algorithm 2
LSBP for Proactive Doppler Shift Compensation Initialize: Set the initial incoming message parameters P (0) j > and µ (0) j can be arbitraryvalue for all j ∈ V and { j, i } ∈ E ; for l ∈ { st , nd , . . . , l thmax } iteration do Vehicle j with j = 1 , · · · , N in parallel Compute (cid:2) C ( l ) j → i (cid:3) − and η ( l ) j → i via (16) and (17) locally at i ; Compute P ( l ) i and µ ( l ) i with (cid:2) C ( l ) j → i (cid:3) − and η ( l ) j → i according to (18) and (19), and ˆ f ( l ) i = µ ( l ) i ; If ∆ i (cid:44) (cid:107) ˆ f ( l ) i − ˆ f ( l − i (cid:107) < th, return current estimate ˆ v ( l ) i ; Transmite P ( l ) i and µ ( l ) i to all its neighbors j ∈ B ( i ) ; end parallel end forend for