Autonomous driving (auto-driving) aims at reducing car accidents, traffic congestion, and greenhouse gas emissions by automating the transportation process [1, 2]. The potential impact of the cross-disciplinary technology has attracted heavy R&D investments not only by leading car manufactures (e.g., Toyota, Tesla, BMW) but also Internet companies (e.g., Google, Apple, Baidu). One primary operation of auto-driving is vehicular positioning [3, 4], namely positioning nearby vehicles and tracking their other parameters such as sizes and trajectories . The information then serves as inputs for computing and control tasks, such as driving-path planning, navigation, and accidence avoidance. The positioning includes both absolute and relative [or called vehicle-to-vehicle (V2V)] positioning and we focus on the latter in this work. One common drawback of existing sensing technologies ranging from RADAR to camera is that they are only capable for detecting the line-of-sight (LoS) vehicles by reflection of light, microwaves, or sound waves but cannot “see through” a large solid object such as a truck, i.e., cannot detect hidden vehicles (HVs) illustrated in Fig. 1. This poses a threat on the safety of auto-driving that was exemplified by numerous recent fatal accidents involving autonomous vehicles. To address this critical issue, this paper presents novel technologies for accurately sensing HV including detecting its position, orientation of driving direction, and size by exploiting cooperative multi-path V2V transmission.
I-a Vehicular Sensing Technologies
Auto-driving requires sensing ranges from tens to hundreds of meters. The current sensing technologies for auto-driving can be separated into passive sensing that includes LIght Detection And Ranging (LIDAR), camera, RAdio Detection And Ranging (RADAR) and cooperative sensing based on wireless message passing.
I-A1 Passive Sensing Technologies
LIDAR, camera, and RADAR share the common feature of detecting the reflections from target vehicles and objects in the environment but using different mediums, namely laser light, natural light, and microwaves, respectively. They are called passive for the reason that the target vehicles and objects are oblivious to the sensing operation.
LIDAR steers ultra-sharp laser beams to scan surrounding environment, analyze the reflected laser signal, and then generate a high resolution three-dimensional (3D) digital map for safe navigation in a dynamic environment 
. On the other hand, a camera can capture the texture, color, and size of a nearby object, facilitating to infer its position by using computer vision techniques. Both LIDAR and camera are inefficient under hostile weather conditions due to the difficulty of light in penetrating fog, snow, and rain  . In addition, LIDAR is currently too expensive to be practical and furthermore the huge amount of generated data via LIDAR or camera is challenging to be processed within the ultra-low latency required for safe driving.
RADAR can localize objects as well as estimate their velocities via sending a designed waveform (e.g., frequency modulated continuous waveform) and analyzing its reflection by the target objects [9, 10]. Particularly, the metal surface of vehicle is capable of reflecting the microwave with negligible absorption that makes RADAR a popular long-range sensing technology for auto-driving. Recent breakthroughs in millimeter-wave (mmWave) RADAR operating in the bandwidth between 76 and 81 GHz makes it feasible deploy large-scale but highly impact arrays for RADAR with sharp beamforming [11, 12]. As a result, the positioning accuracy can be dramatically improved. Moreover, the multiple-input multiple-output (MIMO) RADAR also advances the accuracy-and-latency performance by applying the compressed sensing and sub-Nyquist sampling techniques [13, 14]. Compared with LIDAR and cameras, RADAR can provide longer sensing ranges (up to hundreds-of-meter) and retain the effectiveness under hostile weather conditions or in an environment with poor lighting. However, one main disadvantage of RADAR is that it may incorrectly recognize some harmless small metal object (e.g., a soda can) as a much larger object, leading to false alarms .
Beyond their above relative advantages or disadvantages, the main limitation faced by these passive sensing technologies towards in terms of realizing auto-driving is that they can detect only vehicles and objects with LoS since neither microwaves nor laster light can penetrate a large solid object such as a truck or a building. However, detecting HVs with non-LoS (NLoS) is crucial for collision avoidance in complex scenarios such as overtaking and cross roads as illustrated in Fig. 1.
I-A2 Cooperative Positioning
Cooperative positioning based on V2V or vehicle-to-everything (V2X) communications is a modern approach under active development as part of next-generation intelligent transportation systems . The technology aims at supporting vehicular positioning in environments where Global Positioning System (GPS) is ineffective such as dense urban cities or tunnels . By leveraging V2V communications or V2X connections to the infrastructures, cooperative position allows a vehicle to receive and combine position-related data sent by a group of nearby vehicles, thereby inferring their positions using either statistical inference  or geometric analysis of received signals . Furthermore, 5G mmWave communication techniques can be also applied to further enhance the performance of cooperative positioning .
Nevertheless, the current cooperative technology has its own main drawback. Specifically, its implementation in a dense vehicular network incurs high communication cost and suffers performance degradation due to communication latency, packet loss, limited link life time, inter-vehicle interference among other issues . Furthermore, efficient resource management and multi-access in vehicular networks pose open challenges due to the more complex protocols required for overlaying V2V communications and V2X radio access.
I-B Signal Processing Techniques for Microwave Positioning
While research on LIDAR or camera focuses on mapping, computer vision, and image processing, there exist a rich set of signal processing techniques for microwave positioning, namely a transmitter attempts to estimate the position of a receiver or the reverse. For this purpose, a transmitter transmits a waveform to a receiver, which uses the prior knowledge of the waveform and observes the received signal to estimate the propagation distances, called ranging. Specifically, the receiver estimates the time-of-arrival (ToA) of the received signal if the waveform is a pulse , or the frequency variation if it is a frequency modulated continuous wave (FMCW) . If the transmitter and receiver are synchronized, then ranging can be realized by converting ToA or frequency variation into the propagation distance. Consider a multi-path NLoS channel with scatterers. Provisioned with multi-antenna arrays, the transmitter and receiver are capable of spatial filtering, which allow them to resolve multi-path and estimate the signal’s angles-of-arrival (AoA) and angles-of-departure (AoD) for individual paths besides estimating their distances by ranging . Exploiting the geometric relations between the AoAs, AoDs, and propagation distances enables a transmitter to estimate the position of a receiver or the reverse despite the lack of LoS [25, 26]. One can treat MIMO RADAR as a special implementation of microwave positioning where instead of positioning the transmitter or receiver, they are collocated in a RADAR and collaborate to detect LoS objects (scatterers).
The effectiveness of microwave positioning techniques hinges on the key assumption of perfect synchronization between the transmitter and receiver. This assumption is reasonable for the RADAR application with collocated transmitter and receiver. Another suitable application is mobile positioning using access points where their required synchronization is achieved by a communication protocol . For vehicular sensing, in theory, the application of microwave positioning at a sensing vehicle (SV) should enable it to locate a HV transmitting a suitable waveform. Nevertheless, the required SV-HV synchronization is impractical especially in the presence of high mobility. This is the reason that microwave vehicular sensing is limited to RADAR that is incapable of detecting HVs. Therefore, HV-sensing remains largely an open problem and tackling it is the theme of this work.
I-C Main Contributions
While cameras and LIDAR are suitable for mid-range vehicular sensing (tens of meters), microwave positioning including RADAR can achieve much longer ranges (up to hundreds of meters) as well as coping with hostile weather conditions. Nevertheless, the current technologies for microwave vehicular sensing faces the following key limitations among others.
RADAR, which is widely used for auto-driving, is unable to detect HVs without LoS;
Though there exist microwave positioning techniques for HV detection, their required SV-HV synchronization is difficult to realize in practice due to high mobility and latency criticalness of auto-driving;
Even if the said synchronization is feasible, the existing microwave positioning techniques are incapable of detecting the size and orientation of HV.
In this work, we attempt to address the above issues faced by the state-of-the-art vehicular sensing technologies and thereby advance the area of microwave positioning. To be specific, we present in this paper novel technologies for simultaneously sensing the position, orientation, and size of HVs without requiring SV-HV synchronization. Relaxing the synchronization requirement introduces a design challenge since multi-path propagation distances can no longer be straightforwardly estimated using the conventional ranging techniques. Furthermore, the consideration of HV’s orientation and size introduces new unknowns that complicates the sensing problem. To cope with these challenges, our main design approach is to leverage estimated information on multi-path (including AoA, AoD and ToA) as well the geometry relations between paths so as to construct tractable systems of equations or optimization problems, where the HV position, orientation, and size are unknowns or variables. Then solving them gives the desired HV-state parameters. Based on the approach, a set of HV-sensing techniques are designed for operation in different practical settings ranging from low to high signal-to-noise ratios (SNRs), single-cluster to multi-cluster HV arrays, and small to large waveform sets. The main contributions of this work are summarized as follows.
Sensing HV Position and Orientation: Consider the case where the HV array contains a cluster of collocated antennas. While conventional techniques focus on only positioning, the current design has a more ambitious goal of simultaneously estimating HV’s position and orientation even the absence of SV-HV synchronization. It is assumed that orthogonal waveforms are transmitted over different antennas to enable the SV to estimate the multi-path information (AoA, AoD and ToA). Then given the information and when noise is negligible, a complex system of equations is constructed and solved in sequential steps to obtain the desired HV-state parameters. On the other hand, when noise is present, the sensing problem is suitably formulated as least-square (LS) estimation and also solved in a sequential procedure. It is shown that for accurate HV sensing to be feasible, the required number of paths is at least for 2D propagation or at least for 3D propagation.
Sensing HV Position, Orientation, and Size: Consider the case where the HV array contains multiple clusters of antennas that are distributed over the vehicular body. In this case, the proposed sensing techniques aim at further estimating the HV size along with its position and orientation. Two specific schemes are presented. The first assumes the transmission of multiple orthogonal waveform sets so that the SV can group the paths according to their originating HV antenna clusters. Then the scheme can build on the preceding design in 1) to estimate the HV size via efficiently positioning the HV antenna clusters, which also yields the HV position and orientation. The second assumes the transmission of an identical waveform set such that the said path-grouping is infeasible. Then alternative size detection techniques are proposed based on efficient disk or box minimization under the constraint that the disk or box encloses the HV array. The required numbers of paths for HV-sensing for the first scheme is found to be and that for the second scheme is . Nevertheless, when both of two schemes are feasible, the former outperforms the latter as multiple orthogonal waveform sets help improve sensing accuracy.
Coping with Insufficient Multi-Path: We further propose practical solutions for increasing the number of available paths in the case where they are insufficient for meeting the feasibility requirements of the above HV-sensing techniques. The first solution is to combine paths discovered in multiple time instants by exploiting their randomness in time and the second is to apply random directional beamforming for uncovering hidden paths invisible in the case of isotropic HV transmission. The solutions are complementary and can be jointly implemented to maximize the number of significant paths for enhancing the sensing accuracy.
Realistic Simulation: The proposed HV-sensing techniques are evaluated using practical simulation models of highway and rural scenarios and found to be effective.
The remainder of the paper is organized as follows. Section II introduces the system model and problem formulation for HV-sensing. Sections III and IV present the HV-sensing techniques for the cases of single-cluster and multi-cluster HV arrays, respectively. The solutions for the practical issue of insufficient multi-path for HV-sensing are developed in Section V. Simulation results are presented in Section VI, followed by concluding remarks in Section VII.
Ii System Model
We consider a two-vehicle system where a SV attempts to detect the (relative) position, size, and orientation of a HV blocked by obstacles such as trucks or buildings as illustrated in Fig. 1. An antenna cluster refers to a set of collocated antennas where the half-wavelength antenna spacing is negligible compared with vehicle sizes and propagation distances. An array can comprises a single or multiple antenna clusters, referred to as a single-cluster and a multi-cluster array, respectively. The deployment of a single-cluster array at the HV enables the SV to detect the HV’s position and orientation. On the other hand, a multi-cluster HV array can further make it possible for the SV to estimate the HV size. For the purpose of exposition, in the case of multi-cluster HV array, we consider clusters located at the vertices of a rectangle representing the vehicle. The principle of HV-sensing design in the sequel is based on the efficient detection of the clusters’ positions and thus can be straightforwardly extended to other clusters’ topologies with irregular clusters’ distributions. For HV-sensing in both scenarios, the SV requires only a single-cluster array. Signal propagation is assumed to be contained in the 2D plane and the results are subsequently extended to the 3D propagation. The channel model, V2V transmission, and sensing problem are described in the following sub-sections.
Ii-a Multi-Path NLoS Channel
The NLoS channel between SV and HV contains multi-paths reflected by a set of scatterers. Following the typical assumption for V2V channel, only the received signals at the SV from paths with single-reflection are considered while higher-order reflections are neglected due to severe attenuation . As mentioned, signal propagation is assumed to be constrained within the horizontal plane. Consider a 2D Cartesian coordinate system as illustrated in Fig. 2 where the SV array is located at the origin and the -axis is aligned with the orientation of SV. Further consider a typical antenna cluster at the HV. Each NLoS signal path from the HV antenna cluster to the SV array can be characterized by the following five parameters: the AoA at the SV denoted by ; the AoD at the HV denoted by ; the orientation of the HV denoted by ; and the propagation distance denoted by which is divided into the propagation distance after refection, denoted by , and the remaining distance . The AoD and AoA are defined as azimuth angles relative to orientations of HV and SV, respectively. Fig. 2 graphically shows the definitions of the above parameters.
Ii-B Hidden Vehicle Transmission
To enable sensing at the SV, the HV transmits a set of waveforms defined as follows. Each antenna cluster at HV has antennas with at least half-wavelength spacing between adjacent antennas. Consider a typical antenna cluster. A set of orthogonal waveforms are transmitted over different antennas. Let be the finite-duration baseband waveform in assigned to the -th HV antenna with the bandwidth . Then the waveform orthogonality is specified by with the delta function if and
otherwise. The transmitted waveform vector for the-th HV antenna cluster is . In the case of multi-cluster HV array, the waveform sets for different clusters are either identical or orthogonal with each other. The use of orthogonal waveform sets allows SV to group the detected paths according to their originating antenna clusters as elaborated in the sequel, and hence this case is referred to as decoupled clusters. Then the other case is called coupled clusters. With the prior knowledge of transmitted waveforms, the SV with antennas can scan the received signal due to the HV transmission to resolve multi-path as discussed in the next sub-section.
The expression of the received signal is obtained as follows. Consider a typical HV antenna cluster. Based on the far-field propagation model, the cluster response vector is a function of the AoD defined as
where denotes the carrier frequency and refers to the difference in propagation time to the corresponding scatterer between the -th HV antenna and the first HV antenna in the same cluster, i.e., . Similarly, the response vector of the SV array is written in terms of AoA as
where refers to the difference of propagation time from the scatterer to the -th SV antenna than the first SV antenna. We assume that SV has prior knowledge of both the HV-and-SV array configurations and thereby the response functions and . This is feasible by standardizing the vehicular arrays’ topology. In addition, the Doppler effect is ignored based on the assumption that the Doppler frequency shift is much smaller than the waveform bandwidth and thus does not affect waveform orthogonality.111To be specific, the doppler frequency with the relative velocity of HV to SV and the light speed (meter/sec) is ignored because and the resultant Doppler phase shifts in and are almost zero within the waveform duration. Let denote the index of HV antenna cluster and denote the number of received paths originating from the -th HV antenna cluster. The total number of paths arriving at SV is . Represent the received signal vector at SV as that can be written in terms of , and as
where and denote the complex channel coefficient and ToA of path originating from the -th HV antenna cluster, respectively, and represents channel noise. With no synchronization between the HV and SV, the SV has no information of HV’s transmission timing. Therefore, it is important to note that differs from the corresponding propagation delay, denoted by , with and being the propagation distance. Given an unknown clock-synchronization gap between the HV and SV denoted as , .
Remark 1 (Multi-Access in a Vehicular Network).
Building the above transmission scheme, multi-access techniques can be designed to allow the implementation of the proposed HV-sensing technology in a vehicular network. An example is given as follows. A base station periodically broadcasts the lists of available orthogonal waveform sets, each of which includes different orthogonal waveforms. Then by sensing the ambient signal, each HV can select and transmit one available waveform set unused by nearby vehicles with random timing. Then a SV can decouple the signals from different HVs provided 1) they transmit different waveform sets or 2) their waveforms do not overlap in time upon arrival at the SV which is likely due to bursty transmissions.
Ii-C Estimations of AoA, AoD, and ToA
The sensing techniques in the sequel assume that the SV has the knowledge of AoA, AoD, and ToA of each receive NLoS signal path, say path , denoted by where . The knowledge can be acquired by applying classical parametric estimation techniques briefly described as follows.
Sampling: The received analog signal and the waveform vector are sampled at the Nyquist rate and converted to the digital signal vectors and , respectively.
Matched Filter: The sequence of is matched-filtered by . The resultant coefficient matrix is given by . The sequence of ToAs can be estimated by detecting peak points of the matrix norm , denoted by , which can be converted into time by multiplication with the time resolution . It is worth mentioning that one peak point can contain multiple signal paths if the signals arrive within the same sampling interval.
Multi-Path Estimation: Given , AoAs and AoDs are jointly estimated using a 2D-multiple signal classification (MUSIC) algorithm, which is the most widely used subspace-based detection method . The estimated AoA and AoD are paired with the corresponding estimated ToA , which jointly characterize path .
Ii-D Hidden Vehicle Sensing Problem
The SV attempts to sense the HV’s position, size, and orientation, which can be obtained by using parameters of AoA , AoD , orientation , distances and , and location of multi-cluster HV array. Noting the first two parameters are obtained based on the estimations in Section II-C and the goal is to estimate the remaining parameters.
Iii Sensing Hidden Vehicle with a Single-Cluster Array
In this section, we consider the scenario that a single-cluster array is deployed at HV. Then this section focuses on designing the sensing techniques for SV to detect 1) the HV position (i.e., position of the single-cluster array), specified by the coordinate , and 2) the HV orientation, specified by (see Fig. 2). Based on the path-geometry in Fig. 2, can be described as
The prior knowledge that SV has for sensing is the parameters of NLoS paths estimated as described in Section II-C. Each path, say path , is determined by the parametric set and orientation as (3) shows. Then given the equations in (4), the sensing problem for the current scenario reduces to
The problem is solved in the following subsections.
Iii-a Sensing Feasibility Condition
In this subsection, it is shown that for the HV-sensing to be feasible, there should exist at least four NLoS paths. To this end, by using (3) and multi-path-geometry, we can obtain the following system of equations:
where denotes the coordinate characterized via path . The number of equations in E1 is , and the above system of equations has a unique solution when the dimensions of unknown variables are less than . Since the AoAs and AoDs are known, the number of unknowns is including the propagation distances , , and orientation . To further reduce the number of unknowns, we use the propagation time difference between signal paths also known as TDoAs, denoted by , which can be obtained from the difference of ToAs as where . The propagation distance of signal path , say , is then expressed in terms of and as
Proposition 1 (Sensing Feasibility Condition).
To sense the position and orientation of a HV equipped with a single-cluster array, at least four NLoS signal paths are required: .
Remark 2 (Asynchronization and TDoA).
Recall that one sensing challenge is asynchronization between HV and SV represented by , which is a latent variable we cannot observe explicitly. Considering TDoA helps solve the problem by avoiding the need of considering via exploiting the fact that all NLoS paths experience the same synchronization gap.
Iii-B Hidden Vehicle Sensing without Noise
Consider the case of a high receive signal-to-noise ratio (SNR) where noise can be neglected, i.e., the estimations of AoA/AoD/ToA are perfect. Then the HV-sensing problem in (4) is translated to solve the system of equations in E1. One challenge is that the unknown orientation introduces nonlinear relations, namely and , in the equations. To overcome the difficulty, we adopt the following two-step approach: 1) estimate the correct orientation via its discriminant introduced in the sequel; 2) given , the equation becomes linear and thus can be solved via LS estimator, giving the position . To this end, the equations in E1 can be arranged in a matrix form as
where and . For matrix , we have
with and , and is obtained by replacing all operations in (7) with operations. Next,
and is obtained by replacing all in (9) with .
1) Computing : Note that E2 becomes an over-determined linear system of equations if (see Proposition 1), providing the following discriminant of orientation . Since the equations in (6) are based on the geometry of multi-path propagation and HV orientation as illustrated in Fig. 2, there exists a unique solution for the equations. Then we can obtain from (6) the following result, which is useful for computing .
Proposition 2 (Discriminant of Orientation).
With , the unique exists when is orthogonal to the null column space of denoted by :
Given this discriminant, a simple 1D search can be performed over to find .
Iii-C Hidden Vehicle Sensing with Noise
In the presence of significant channel noise, the estimated AoAs/AoDs/ToAs contain errors. Consequently, HV-sensing is based on the noisy versions of matrix and , denoted by and , which do not satisfy the equations in E2 and (10). To overcome the difficulty, we develop a sensing technique by converting the equations into minimization problems whose solutions are robust against noise.
1) Computing : Based on (10), we formulate the following problem to find the orientation :
Solving the problem relies on a 1D search over .
2) Computing : Next, given , the optimal can be derived by using the LS estimator that minimizes the squared Euclidean distance as
which has the same structure as (11). Last, the origins of all paths can be computed using the parameters as illustrated in E1. Averaging these origins gives the estimate of the HV position with and .
Iii-D Extension to 3D Propagation
Consider the scenario that propagation paths lie in the 3D Euclidean space instead of the 2D plane previously assumed. As shown in Fig. 3, the main differences from the 2D scenario are that the elevation angles are added to the AoAs, AoDs, and HV orientation. Specially, the AoA includes two angles: (azimuth) and (elevation) and AoD consists (azimuth) and (elevation). The estimations of AoAs and AoDs in the 3D model can be jointly estimated via various approaches, e.g., MUSIC algorithm for 3D signal detection (see e.g., ). The HV orientation also includes two unknowns: (azimuth) and (elevation). The coordinates of HV, denoted by , are given as
where . Then, similar to E1, the following system of equations is constructed for 3D propagation:
where . It is shown that the number of equations and the number of unknown variables are and , respectively. For the HV-sensing problem to be solvable, we require , which leads to the following proposition.
Proposition 3 (Sensing Feasibility Condition for 3D).
Consider the 3D propagation model. To sense the position and orientation of HV provisioned with a single-cluster array, at least three NLoS signal paths are required, i.e., .
Compared with 2D propagation, the minimal number of required signal paths is reduced because extra information can be extracted from one additional dimension (i.e., elevation angles information of AoAs, AoDs) of each signal path. A similar methodology described in Sections III-B and III-C can be easily modified for 3D propagation by applying a 2D search based discriminant to find and over and , respectively. The details are omitted for brevity.
Iv Sensing Hidden Vehicle with a Multi-Cluster Array
The preceding section targets the scenario that the HV is provisioned with a single-cluster array, allowing the SV to sense the HV position and orientation. In this section, we consider the scenario where a multi-cluster array is deployed at HV so that SV can sense HV’s array size (approximating the HV size) in addition to its position and orientation. Sensing techniques are designed separately for two cases, namely decoupled and coupled HV antenna clusters, in the following sub-sections.
Iv-a Case 1: Decoupled HV Antenna Clusters
Consider the case of decoupled HV antenna clusters via transmission of orthogonal waveform sets over different clusters. As a result, the SV is capable of grouping detected paths according to their originating clusters. This simplifies the HV-sensing in the sequel by building on the techniques in the preceding section.
Recall that four HV antenna clusters are located at the vertices of a rectangle with length and width that represents the HV shape (see Fig. 4). The vertex locations are represented as . Different orthogonal waveform set is assigned to each cluster, allowing SV with prior knowledge on the waveform sets to differentiate the signals transmitted by different clusters. The more challenging case where all clusters are assigned an identical waveform set is studied in the next section. Let each path be ordered based on HV array index such that where represents the set of received signals from the -th HV antenna cluster. Note that the vertices determines the HV size and their centroid that gives the HV position. Therefore, the sensing problem can be represented as
A naive sensing approach is to exploit the orthogonality of multiple waveform sets to decompose the sensing problem into separate positioning of HV antenna clusters using the technique designed in the preceding section. In the following subsection, we propose a more efficient sensing technique exploiting the prior knowledge of the HV clusters’ rectangular topology.
Iv-A1 Sensing Feasibility Condition
Without loss of generality, assume that the received signal from the first HV antenna cluster, indexed by the set , is not empty and . Based on the rectangular configuration of (see Fig. 4), a system of equations is formed:
and is obtained via replacing all and in with and , respectively. The number of signal paths is given as . Compared with E1, the number of equations in E4 is the same as while the number of unknowns increases from to since and are also unknown. Consequently, E4 has a unique solution when .
Proposition 4 (Sensing Feasibility Condition).
Consider the scenario that the HV is provisioned with a -cluster array and orthogonal waveform sets are transmitted from different clusters. To sense the position, size, and orientation of the HV, at least six paths are required: .
Remark 3 (Advantage of Array-Topology Knowledge).
The separate positioning of individual HV antenna clusters requires at least NLoS paths (see Proposition 1). On the other hand, the proposed sensing technique reduces the number of required paths to only by exploiting the prior knowledge of the rectangular configuration of antenna clusters.
Iv-A2 Hidden Vehicle Sensing
Consider the case where channel noise is negligible. The system of equations in E4 can be rewritten in a compact matrix form:
where with following the index ordering of , and is given in (8). The matrix can be decomposed as
where follows (6). Moreover, is given as with
where counts the number of elements in and is obtained by replacing all in with . Similarly, can be written as where
and is obtained by replacing all in with .
Proposition 5 (Discriminant of Orientation).
With , the unique exists when is orthogonal to the null column space of denoted by :
Given this discriminant, a simple 1D search can be performed over the range to find .
2) Computing : Given the , E5 can be solved by
Iv-B Case 2: Coupled HV Antenna Clusters
It is desired to reduce the number of orthogonal waveform sets used by a HV so as to facilitate multi-access by dense HVs. Thus, in this section, we consider the resource-limited case of coupled HV antenna clusters where a identical waveform set is shared and transmitted by all HV antenna clusters. The design of HV-sensing is more challenging since the SV is incapable of grouping the signal paths according to their originating HV antenna clusters. For tractability, the objectives of HV-sensing for this scenario is redefined as: 1) positioning of the centroid of HV multi-cluster array denoted by ; 2) sensing the HV size by estimating the maximum distance between HV antenna clusters and , denoted by ; 3) estimating the HV orientation . It follows that the sensing problem can be formulated as
To solve the problem, we adopt the following two-step approach:
Step 1: By assuming that all signals received at SV originate from the same transmitting location, it is treated as the HV array centroid and estimated together with the orientation using the technique in Section III.
Step 2: Given and , the size parameter can be estimated by solving optimization problems based on bounding the HV array by either a disk or a box.
The techniques for Step 2 are designed in following sub-sections.
Iv-B1 HV Size Sensing by Disk Minimization
To the purpose of algorithmic design, the HV array is outer bounded by a disk. Then the problem of estimating the HV size parameter at SV can be translated into the optimization problem of minimizing the bounding-disk radius. As shown in Fig. 5(a), we define a sensing disk centered at the estimated centroid with the radius :
A constraint is applied that all HV antennas, or equivalently the origins of all signal paths received at the SV, should lie within the disk. Then estimating the HV size can be translated into the following problem of disk minimization:
where the first constraint is as mentioned above and the second represents the distance after the reflection cannot exceed the total propagation distance represented in terms of and TDoA as with being the TDoAs [see (5)]. One can observe that Problem E6 is a problem of second-order cone programming (SOCP). Thus, it is a convex optimization problem and can be efficiently solved numerically e.g., using the efficient MatLab toolbox such as CVX.
Analyzing the problem structure can shed light on the number of required paths for HV-sensing in the current scenario. The existence and uniqueness of the optimal solution for Problem E6 can be explained intuitively by considering the feasible range of the optimization variable . Let represent the feasible range of given the disk radius by considering only path :
Then the feasible range of , denoted by , can be written as It is straightforward to show the following monotonicity of : if with . Based on the monotonicity, one can conclude that there always exists an optimal and unique solution for Problem E6 such that if or otherwise . In other words,
The value corresponds to the critical case where there exist two feasible range sets and only contact each other at their boundaries such that contains a single feasible point that corresponds to . This leads to the following proposition.
Proposition 6 (HV Size Sensing by Disk Minimization).
Given the solution for Problem E6, there always exist at least two paths, say and , whose originating positions lie on the boundary of the minimized disk :
Remark 4 (Feasible Condition of HV Sensing by Disk Minimization).
Remark 5 (Extension to 3D Propagation).
The extension to 3D propagation model in Section III-D is straightforward by using a sphere instead of a disk [see Fig. 5(b)]. The resultant sphere minimization problem has the same form as Problem E6 except that the first constraint modified as
where the centroid is estimated using the technique in Section III-D. Again, the problem can be optimally solved since it still follows SOCP structure.
Iv-B2 HV Size Sensing by Box Minimization
In the preceding sub-section, the HV size is estimated by bounding the HV array by a disk and then minimizing it. In this sub-section, the disk is replaced by a box (rectangle) and the HV size estimation is translated into the problem of box minimization. Compared with disk minimization, the current technique improves the estimation accuracy since a vehicle typically has a rectangular shape. Let and denote the length and width of the rectangular where the HV antenna clusters are placed at its vertices [see Fig. 5(c)]. Then the problem of HV size sensing is to estimate both and . Recall that the HV array centroid and orientation are estimated in Step 1 of the proposed sensing approach as mentioned. Given and , we define a sensing box for bounding the HV array, denoted as , as an -rotated rectangle centered at and having the length and width :
where is the counterclockwise rotation matrix with the rotation angle given as
and represents an element-wise inequality. Like disk minimization in the previous subsection, finding the correct and is transformed into the following box minimization problem:
Problem E7 can be solved by quadratic programming (QP), which is a convex optimization problem and can be efficiently solved using a software toolbox such as MatLab CVX. A result similar to that in Proposition 6 can be obtained for HV size sensing by box minimization as shown below.
Proposition 7 (HV Size Sensing by Box Minimization).
Given the solution for Problem E7, there always exist at least two paths, say and , whose originating positions lie on two different vertices of the minimized box:
Proof: See Appendix -B.
Remark 6 (Feasible Condition of HV-Sensing by Box Minimization).
A similar remark as Remark 4 for disk minimization also applies to the current technique. Specifically, though two paths are required to determine the optimal box length and width based on Proposition 7, at least four paths are required for estimating the required HV centroid and orientation (see Proposition 1).
Remark 7 (Sensing Box Minimization for Decoupled Antenna Clusters).
The technique of HV size sensing by box minimization developed for the case of coupled HV antenna clusters can be also modified for use in the case of decoupled clusters. Roughly speaking, the modified technique involves separation minimization of four boxes corresponding to the positioning of four antenna clusters. As the modification is straightforward, the details are omitted for brevity. The resultant advantage with respect to the original sensing technique proposed in Section IV-A is to reduce the minimum number of required paths from (see Proposition 4) to .
Remark 8 (Extension to 3D Propagation).
Similar to Remark 5 for disk minimization, the technique of HV size sensing by box minimization originally designed for 2D propagation can be extended to 3D propagation model by using a cuboid instead of a box, yielding the problem of cuboid minimization as illustrated in Fig. 5(d). Compared with E7, the objective function of the cuboid minimization is where the new variable is added to represent the height of the cuboid. In addition, the first constraint in E7 is modified as
where is the 3D counterclockwise rotation matrix with the rotation angles and as
and the centroid can be obtained by the technique in Section III-D. The problem of cuboid minimization is still QP and the solution approach is similar to that for the 2D counterpart.
V Coping with Insufficient Multi-Path
The HV-sensing techniques designed in the preceding sections require at least four propagation paths to be effective. In practice, it is possible that the number of received paths may be insufficient, i.e., , due to either sparse scatterers or that most paths are severely attenuated by multiple reflections. To address this practical issue, two solutions are proposed in the following sub-sections, called sequential path combining and random directional beamforming. For simplicity, we focus on the case of single-cluster HV array while the extension to the case of multi-cluster array is straightforward.
V-a Sequential Path Combining
As shown in Fig. 6, the technique of sequential path combining implemented at the SV merges paths from repeated transmissions of HV till a sufficient number of paths is identified for the purpose of subsequent HV-sensing. Let denote the number of HV’s repetitive transmissions with a constant interval denoted by . The interval is chosen to be much larger than the delay-spread of each transmission, enabling SV to differentiate the arrival paths according to their transmission time instants. Let and denote the time instant of the -th transmission and the corresponding set of detected paths, respectively. Assume that the relative orientation of driving direction and velocity of HV with respect to SV, namely and , remain constant within the entire duration of intervals . Then the following system of equations are formed: