Signalling Design in Sensor-Assisted mmWave Communications for Cooperative Driving

Millimeter-Wave (mmWave) Vehicle-To-Vehicle (V2V) communications are a key enabler for connected and automated vehicles, as they support the low-latency exchange of control signals and high-resolution imaging data for maneuvering coordination. The employment of mmWave V2V communications calls for Beam Alignment and Tracking (BAT) procedures to ensure that the antenna beams are properly steered during motion. The conventional beam sweeping approach is known to be unsuited for the high vehicular mobility and its large overhead reduces transmission efficiency. A promising solution to reduce BAT signalling foresees the integration of V2V communication systems with on-board vehicle sensors. We focus on a cooperative sensor-assisted architecture for mmWave V2V communications in line of sight, where vehicles exchange the estimate of antenna position and its uncertainty to compute the optimal beam direction and dimension. We analyze and compare different signalling strategies for sharing the information on the antenna estimate, evaluating the tradeoff between signalling overhead and performance loss for different position and uncertainty encoding strategies. Main attention is given to differential quantization on both the antenna position and uncertainty. Analyses over realistic urban mobility trajectories suggest that differential approaches introduce a negligible performance loss while significantly reducing the BAT signalling communication overhead.

best Tx and Rx beam pair [11]. This procedure is extremely time demanding, especially when a high number of beam pairs needs to be tested, and it is clearly unfeasible with vehicles travelling at high speed or in trajectories including sharp turns and obstacle avoidance maneuvers. Some first approaches have been proposed to fasten the alignment procedure [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24]. Authors of [12], [13], [14] leverage on vehicle traffic information to select the beam pairs; prior knowledge, however, may not be always available. The analysis proposed in [25] suggests the use of Vehicle-To-Infrastructure (V2I) communications to manage the performance decay in terms of packet loss, delay and throughput caused by an increase of CAVs penetration in the scenario. Works in [17], [18], [19] exploit a radar placed on a Road Side Unit (RSU) to track vehicles' motion and accordingly update the beam pointing direction. A similar approach is used in [20], [21] to integrate the motion control of an autonomous vehicle with beam steering, to provide a stable connection with a base station. RSUs are employed for BW and tracking also in [26], where vehicles are divided in groups based on their distance and motion direction. RSUs with imaging capabilities are considered also in [27], [28], which exploit camera and a lidar measurements, respectively, to define the pointing angles of a vehicular network.
V2I-based solutions require the deployment of a dedicated infrastructure and they do not focus on the more challenging Vehicle-To-Vehicle (V2V) communication, where both Tx and Rx are moving. The work in [29] includes the design of a position-assisted V2V link assuming the presence of a reconfigurable intelligent surface. Authors of [30] propose two beam alignment strategies, either refining an initial wide beam, or continuously testing a subset of narrow beams. Rather than checking several pointing directions, other approaches foresee a tight integration of vehicle navigation and communication. Authors of [15], [16] propose to support beam alignment by coupling mmWave with sub-6 GHz V2V communications. In particular, in [16] information concerning the vehicle state (i.e., location and speed) is transmitted in the sub-6 GHz beacons and used by the vehicles to schedule the transmission and select the beam. Lastly, the work in [24] suggests the use of city map or traffic pattern, together with vehicle position estimate, to optimize the sweeping over a non-uniform set of beams.
As main outcome, all the above cited works highlight the importance of an accurate knowledge of the vehicle state in terms of position and, possibly, orientation [31]. Considering the high diffusion of on-board sensors on commercially available vehicles [32], [33] and the increasing trend of automation, exploiting such hardware to provide position information at the PHY layer of the communication appears as the most viable and cost-effective solution. In this direction, the work in [22] proposes to firstly estimate the beam pointing from vehicle positioning sensors and then rely on velocity information to perform beam tracking.
The authors of [34], on the other hand, propose to use onboard cameras to select beam pairs and determine the relative coherence time. Cameras are paired with lidar in [35] for detecting and tracking the target vehicle of the V2V communication, but no explicit analysis on the V2V link is included. Beam pointing and tracking can also be driven by inertial sensors mounted at the antenna [36], [37]. The accuracy of sensors' data, however, is still limited by bias that severely affects the overall communication performance. To enhance the quality of the localization, authors of [23] implement a data fusion algorithm that reduces the uncertainty of the vehicle position information and design a cooperative BAT scheme between the communicating vehicles. The results therein included only refer to a simulation of a highway overtaking scenario, reducing the generality of the approach. Authors of [5], [38] use positioning and visual measurements to predict the motion of the receiver and track the beam direction. In [39] location measurements are used to reduce the number of tested beam pairs and speed up the alignment procedure: sensors data are processed to create an area where the Rx is expected to lie and only the beams crossing this area are tested, resulting in a lower overhead. The work in [40] follows a similar approach, adding the computation of an optimal BW to maximize the channel capacity. Enlarging or reducing the beam dimension has been investigated also in [41], [42], [43], [44], [45] where the authors focus on the V2I case. The adaptation of the BW in V2V beam alignment scenario, instead, is considered in [46], where channel and queue state information of the link are exploited. The work in [47] proposes to compute the optimal beam as a function of the mean data-rate. However, the authors do not take into consideration the localization uncertainty for the Rx vehicle (only for the Tx one). Authors of [48] provide a location-based beam alignment and adaptation to optimize the inter-vehicular interference. That work, however, takes into consideration only Global Positioning System (GPS) measurements, without exploiting additional sensors.
We propose a new strategy that allows to exploit the available location information from on-board equipment, combining it with V2V cooperation and data fusion to reduce the uncertainty on the estimate of vehicle state (position and orientation), along with an optimization of the required signalling, thus minimizing the communication overhead. This work is rooted in [49], [50] for what concerns the cooperative Sensor-Assisted (SA) architecture for BAT, whereas the signalling analysis in a 3GPP C-V2X standard context and related optimization is a new original contribution of this paper. We design a parameter exchange strategy to support a cooperative BAT between a pair of vehicles in Line Of Sight (LOS) conditions. In particular, we study different information quantization strategies to disseminate i) the antenna position estimate and ii) associated uncertainty (i.e., variance). We propose an encoder fully compliant with the existing standards [6], [51], and a differential approach which leverages on the variation of the considered quantities over a communication frame that reduces the signalling overhead.
The proposed SA methodology requires the information of on-board sensors to be available at the PHY layer of V2X communication to implement a BAT procedure. Positionrelated information is nowadays available at the facility layer of the V2X protocol stack, as it is conveyed, for instance, into Cooperative Awareness Messages (CAM). However, the current dissemination frequency of CAM is prescribed by the current standard up to 10 Hz [52], which does not match the update rate required by a beam steering in high mobility systems. As an example, let us consider two connected vehicles, namely v 1 and v 2 , both equipped with an Uniform Planar Array (UPA). Suppose v 1 is transmitting to v 2 with a BW φ. The inter-vehicle distance is d = 20 m. Assume v 2 is moving in a direction orthogonal to the LOS between v 1 and v 2 with speed ν = 40 km/h. If the pointing direction of vehicle v 1 does not change, the time T required by vehicle v 2 to exit the coverage area of the beam is T = d tan(φ)/ν. If the UPA has 8 antennas, i.e., φ = 12.74 deg, this time is T = 120 ms. If the UPA has twice the number of antennas (φ = 6.37 deg), the time reduces to T = 60 ms. This shows that the current CAM update frequency results unsuited for a cooperative beam alignment scheme when extremely narrow beams are employed. To overcome this limitation, we here propose position-related information to be exchanged at the PHY level of a C-V2X protocol. This is not allowed in current V2X communication standard [6], [53], which does not allocate space for such information, meaning that no integration of telecommunication and sensor information is considered at time being. Once the two vehicles are connected, in fact, refinement of Tx and Rx beams are performed through transmission of channel state information messages and sounding reference signals. The latest release of 3GPP C-V2X, namely Rel. 17 [54], introduced a number of novelties (spectrum expansion up to 71 GHz, enhancement of mMIMO system and expansion of sidelink supported band) but signalling of position information at PHY is not considered yet. We show in this work that a modification in this sense would allow for a robust BAT procedure at the cost of a limited overhead and, most notably, would improve the throughput of the V2V link.
The remainder of this paper is organized as follows: Section II describes the V2V channel, SA architecture and the communication hardware to enable BW adaptation. Section III includes the design of signalling strategies of position and covariance information in the V2V link to support SA method. Section IV describes the urban traffic simulations with Simulator of Urban MObility (SUMO) software and the numerical results. Finally Section V draws conclusions of this work.

II. SYSTEM MODEL
This section is devoted to describe the SA architecture, the MIMO antenna models and the performance metrics that will be used to evaluate the designed signalling solutions.

A. SENSOR-ASSISTED BEAM CONTROL
Let us consider a three dimensional (3D) Cartesian Navigation Reference System (NRS), with axes (X, Y, Z), to model the dynamics of the V2V link between two vehicles (v 1 and v 2 ), as shown in Fig. 1 contains the coordinates of the Centre of Gravity (COG) of vehicle v at time t, whereas q v (t) ∈ R 4 is the quaternion parametrizing vehicle's orientation [55], both referred to NRS. The antenna position is In our study, the antennas are placed on the vehicle's roof, according to the 3GPP recommendations in [56]. Assuming the vehicle is a rigid body, the following holds [57]: where r v ∈ R 3 is a constant vector defining the distance of the antenna with respect to the COG and R(q v (t)) ∈ R 3×3 is a rotation matrix that accounts for the instantaneous vehicle orientation q v (t), as detailed in [55, eq. (A10)]. We consider the SA architecture in [50], where each vehicle is equipped with a GPS and an Inertial Measurement Unit (IMU). The observations provided by GPS and IMU sensors are fused by means of an Extended Kalman Filter (EKF), designed as in [55] for tracking the vehicle dynamics. The outputs of the EKF are an estimate of vehicle position and orientation, p v (t) and q v (t), respectively, and the related covariance matrices C Since r v is deterministic (a vehicle-dependent parameter), the second term in (2) accounts only for the uncertainty on the orientation that typically has a negligible impact with respect to the first one. It follows that C a v (t) ≈ C p v (t). Let us consider the perspective of vehicle v 1 , which knows and receives a 2 (t) and C a 2 (t) from VOLUME 4, 2023 495 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
vehicle v 2 . The following disclosure is also valid for v 2 perspective, with proper adjustments. The LOS direction is determined by the Tx and Rx antenna positions. In the ideal case of perfect mutual awareness of the positions, the azimuth and elevation pointing angles, α 1 (t) and β 1 (t) respectively, are The pointing directions are collected into vector ξ 1 Since v 1 knows a 1 (t) and a 2 (t), it computes the estimate of the pointing directions ξ 1 (t), by means of (3). The considered SA procedure assumes that v 1 is capable of enlarging or reducing the BW according to the quality of the beam tracking. Intuitively, the more accurate is the estimate of the state of both vehicles, the narrower can be the beam. BW adaptation is performed as described in [50] and it considers the covariance of ξ 1 (t) as an indicator of the reliability of the LOS estimate. The covariance of the LOS estimate available at vehicle v 1 The standard deviations of azimuth and elevation estimates, namely σ α 1 (t) and σ β 1 (t), are computed as where b α,1 (t) ∈ R 3×1 and b β,1 (t) ∈ R 3×1 are the gradients of α 1 (t) and β 1 (t) with respect to a 1 1,2 (t). The azimuth and elevation standard deviations σ α 1 (t) and σ β 1 (t) are used to compute the azimuth and elevation components of optimal which cover the ±kσ confidence interval of pointing errors (in this work k = 3 to account for 99.7% of cases).

B. MIMO ANTENNA
Each vehicle is assumed to be equipped with two MIMO UPAs, both placed on the roof, one at the front and one at the rear, according to [56]. To simplify the reasoning, the vehicle has the possibility to select one of them at a time, according to the pointing direction. The UPA has N tot ant × N tot ant antenna elements overall, half-wavelength spaced. To shape a communication beam with an optimized dynamic BW, we ). An example of BW control is given in [58]. Assuming a circular shaped beam, the approach we employ to adapt the BW consists in turning on/off the antennas so that N on ant,v (t) is the same along the horizontal and vertical array dimension. This introduces the constraint i.e., the BW is the same along the azimuth and elevation directions. Considering that the motion of the vehicles mostly happens on the horizontal plane (XY of the NRS), we have The relation between the number of active antennas N on ant,v (t) and φ v (t) is as follow [59] φ where the term cos (α v (t)) denotes a broadening of φ v (t) according to the pointing direction. From (7), the number of active antennas is

C. PERFORMANCE METRICS
Moving from the ideal case of Section II-A to a practical V2X system, the knowledge of pointing angles is unavoidably subject to an error This pointing mismatch reduces the antenna gain with respect to the ideal case (i.e., ξ v = [0 0] T ) by a quantity that is computed as in the following. Let the antenna gain associated to the estimated pointing angles be where s((ξ v (t)+ ξ v (t))) and s(ξ v (t)) are the steering vectors corresponding to the pointing direction . . .
being μ i ∈ R 3 the position vector of the i-th element of the antenna and w( ] T the wave propagation direction along the LOS. In case of perfect pointing, i.e., is the ideal gain as function of ξ v (t).
In our analysis we will use the mismatched antenna gain to compute the SNR of the link and thus to assess the performance loss due to pointing errors. Notice that, in the following, we are setting v 1 and v 2 as Tx and Rx respectively, but same procedure applies to the reverse link. The SNR is written, in dB scale, as where , P 1 is the Tx power, P n is the noise power at each element of the receiving antenna array of vehicle v 2 , d(t) is the distance between the vehicles and PL(d(t)) represents the path loss. The latter is [61] PL dB (d(t)) = −147.55 + 10 η log 10 (d(t)) + 20 log 10 (f ), (12) where f is the carrier frequency and η is the pathloss exponent. P dB n is computed as being NF dB the noise floor and B the communication bandwidth.
In case of perfect beam pointing, the ideal SNR is where G dB id (ξ 1 (t), ξ 2 (t)) = G id,1 (ξ 1 (t)) + G id,2 (ξ 2 (t)). To highlight only the losses associated to misalignment, we define the SNR loss as in (15), shown at the bottom of the page, where the terms related to path loss and noise floor cancel out, since they are equal for both SNR dB (t) and SNR dB id (t). The last two metrics used for performance assessment are i) the fading duration F, defined as the time interval in which the SNR persists in outage, i.e., SNR(t) < SNR, with SNR the outage threshold; ii) channel capacity C defined as according to Shannon-Hartley theorem. The adopted metrics focus on the evaluation of the performance degradation caused only by incorrect beam alignment. For this reason, the V2V communication channel is assumed as ideal, neglecting the effects of multipath propagation and encoding/decoding stages.

III. V2V SIGNALLING DESIGN
In this section, we propose multiple strategies to optimize the signalling of the information required by the SA approach with adaptive BW between a pair of connected vehicles, v 1 and v 2 , as described above in Section II-A. Recall that the information to be shared by vehicle v is the estimate of the antenna position a v ∈ R 3×1 and the related covariance matrix C a v (t) ∈ R 3×3 . For the sake of brevity, we describe the signalling from the perspective of vehicle v 1 , which receives a 2 (t) and C a 2 (t) every τ = 10 ms (the duration of a 5G frame [6]). Note that only the quantization and signalling of these quantities related to vehicle v 2 is derived, since a 1 (t) and C a 1 (t) are assumed to be available at vehicle v 1 at each time instant.

A. SIGNALLING OF POSITION INFORMATION
The dissemination of position information by v 2 allows v 1 to compute ξ 1 (t) from (3). For this task, we design two different approaches. In the first one the vehicles share their absolute position estimate, in the second one they only send the variation of the position with respect to the latest available information, with the scope of reducing the signalling overhead.

1) ABSOLUTE POSITION SIGNALLING
The first proposed approach assumes to share a 2 (t), i.e., the antenna position estimate with respect to an absolute spatial reference system. To this aim we employ the WGS84 reference system [62], the standard for encoding absolute position of GPS, which uses N pos bit = 32 bits for each of the SNR dB loss (t) = P dB

2) DIFFERENTIAL POSITION SIGNALLING
To reduce the signalling rate, the second designed approach proposes a differential encoding of position. Consider two consecutive signalling instants, t −τ and t, and the estimated antenna position vectors of vehicle v 2 in such time instants, i.e., a 2 (t−τ ) and a 2 (t). We propose to signal the variation of the antenna position observed over the elapsed time interval, which we refer to as the differential antenna position δ a 2 (t), defined as This approach leverages on the fact that, once the V2V link is established, v 1 can retrieve the position of the antenna of v 2 at time t based on the last available antenna information, i.e., a 2 (t) = a 2 (t − τ ) + δ a 2 (t). Note that, in this case, it is necessary to define a suited quantization interval for signalling δ a 2 (t). To this extent, we consider that current 5G standard [6] is built upon the operability assumption of having vehicles travelling up to ν max = 250 km/h. This means that each component of δ a 2 (t) is assumed to vary less than ν max τ = 0.79 m. We thus conservatively set to ±1 m the quantization interval for the elements of δ a 2 (t). The choice of quantization bits N δpos bit is a tradeoff between the additional error caused by approximation and the communication signalling overhead.
To avoid possible cumulation of errors due to differential encoding, we introduce a periodic signalling of the absolute position a 2 (t) (encoded in n · N pos bit bits) after a number of instances of signalling of δ a 2 (t). Specifically, being τ the periodicity for δ a 2 (t), we indicate with T the periodicity for a 2 (t) set. We set T = 1 s. The required signalling data rate for the differential position approach is where the first term refers to the signalling of δ a 2 , whereas the second one to the signalling of a 2 (t). According to (19), for N δpos bit ∈ {3, 6, 10} bit and recalling that N pos bit = 32, it results that R diff pos ∈ {1, 1.9, 3.1} kbps, respectively. Fig. 3b shows the signalling diagram of the differential position information.

B. SIGNALLING OF LOCATION COVARIANCE INFORMATION
This section discusses the procedures used by vehicle v 2 to share the covariance of antenna position estimate, denoted as Considering the matrix symmetry, univocal representation requires 6 elements. We approximate C a v (t) as a diagonal matrix, assuming that the three position components are uncorrelated, reducing the required elements to n = 3.

1) FULL COVARIANCE SIGNALLING
To quantize the values of C a 2 (t), we rely on the 16 confidence values defined by the SAE standard in [51]. The interval of position confidence values ranges from 0.01 m to 500 m according to a non-uniform quantization. Considering that for beam-based V2V communications it is not attractive to consider the upper and lower values, we limit the confidence values to eight levels, as reported in Table 1, thus requiring N cov bit = 3 bits for each diagonal entry. The rationale behind our choice is that confidence values larger than 20 m are useless for an accurate beam pointing with narrow beam, and values lower than 10 cm are not achievable for current vehicle navigation and tracking algorithms. Confidence values are then squared, to retrieve the values of variance of C a v (t) reported in the third column of Tab. 1. The covariance signalling data rate of this approach with τ periodicity is Fig. 3c shows the diagram for the exchange of full covariance information.

2) DIFFERENTIAL COVARIANCE SIGNALLING
Similarly to the position case, to reduce the signalling overhead we propose to quantize the differential covariance as where diag(·) operator extracts the diagonal elements of a square matrix as for (20). Each entry of δC a 2 (t) is encoded with N δcov bit = 2 bits, indicating a relative jump on the quantization levels in Tab. 1 with respect to the previously available information. Similarly, to avoid error cumulation, periodic exchange every T of the full covariance information (encoded in n · N cov bit bits) is introduced. The required signalling data rate for the differential covariance approach is Fig. 3d shows the signalling diagram for differential covariance exchange.

3) SCALAR COVARIANCE SIGNALLING
As final approach, we propose a method which reduces the covariance information to a single scalar value. The underlying motivation is on the higher uncertainty on the plane where vehicle motion occurs (i.e., σ α v (t) > σ β v (t)). This condition allows to signal only the covariance of the LOS azimuth σ ⊥ 2 (t), which is the component of C a 2 (t) orthogonal to the azimuth LOS direction, as shown in Fig. 4. Denoting the versor of the direction orthogonal to the azimuth as Quantization of σ ⊥ 2 (t) is the same of the differential covariance, for this reason also in this case we introduce the exchange of C a 2 (t) with a higher periodicity T. For this last method, the required signalling data rate is To conclude this section on the designed signalling strategies to support SA architecture, we provide in Tab. 2 a summary of the required data rate for all presented methods, computed according to (17), (19), (21) and (23) and considering N δpos bit = 6 bit, τ = 10 ms and T = 1 s. Notice that methods that do not allow adaptation of the BW do not include covariance information exchange.

IV. SIMULATIONS AND RESULTS
This section presents an analysis of the performance of the designed SA signalling strategies, to evaluate the impact of the information encoding (i.e., location and related accuracy) on V2V communication, in terms of link quality (i.e., the SNR loss due to quantization), transmission capacity and robustness to fading. The simulations are performed using realistic planar trajectories generated with Simulator of Urban MObility (SUMO) software, considering an urban area in Milan, Italy, and the framework described in [63] to model the effect of road roughness on vehicle position and orientation, according to ISO 8608 [64]. The resulting 3D vehicle dynamics affect the time evolution of antennas' positions and orientations, which are used to simulate the V2V link.

A. SIMULATION SETTINGS
The simulated urban environment includes 61 pairs of vehicles with trajectories around the Leonardo campus of the Politecnico di Milano, as shown in Fig. 5. Each pair of V2V connected vehicles is simulated with a headway of 5 s between the leader and the follower. With this gap the inter vehicular distance ranges from 10 m to 80 m, with mean value of 37 m. The speed of the vehicles is comprised between 0 km/h and 61.2 km/h, with an average value of 28.8 km/h. The connected vehicles perform beam alignment with the SA architecture described in Section II-A, exchanging their position estimate and associated accuracy with periodicity τ . The optimal BW, computed as in (6), determines the number of active antennas, by means of (8). Then, each vehicle keeps the pointing angles and the BW constant, until it receives a new position and accuracy information (after a period τ ).
The communication parameters f , B, NF dB and the latency are compliant with the 5G V2X standard, whereas η = 2 assumes free space propagation. Notice that we are considering the largest bandwidth prescribed by the standard for mmWave, i.e., B = 400 MHz [6]. The latency quantifies the time interval between the transmission and reception of the position and covariance information among two connected vehicles. The dimensions of vehicles and the performance of on-board sensors are instead consistent with commercially available private vehicles. All the parameters are summarized in Tab. 3.
To show the benefits brought by the proposed signalling strategies, a comparison with a fixed BW BAT procedure relying on currently-available CAM is considered, where vehicles broadcast the ego-antenna position estimate every τ CAM = 100 ms [65]. The position information is encoded according to the WGS84 standard [66], hence the CAMbased BAT is equivalent to the absolute position signalling with τ CAM periodicity.

B. IMPACT OF POSITION QUANTIZATION
The first assessment investigates the effect of position information quantization on the SA communication performances using the signalling strategies described in Section III-A. We employ the differential position approach and evaluate its performance with respect to the absolute position one. Three different values of N pos bit are considered for differential position encoding. To focus on the impact of position quantization, the first analysis considers a timeinvariant number of active antennas N on ant,v , i.e., fixed BW is considered without sharing of covariance information. In Fig. 6 we show the Empirical Cumulative Distribution Function (ECDF) of the SNR loss (15) for N on ant,v = 8 (Fig. 6a) and N on ant,v = 32 (Fig. 6b), which correspond to a fixed BW of φ v = 12.8 deg and φ v = 3.2 deg at broadside, respectively. Comparing the two figures, it emerges that CAM-based position dissemination is not suited to support V2V communications over narrow beams. For N on ant,v = 8 (Fig. 6a) and N on ant,v = 32 (Fig. 6b) the SNR loss is higher than the one of SA methods that employ at least N pos bit = 6 for differential position encoding. The reason for this, as explained by the example in Section I, is the low dissemination rate (10 Hz) of CAM, which is unsuited for highly-dynamic V2V scenarios. Focusing on the proposed method, instead, it can be seen that the additional losses introduced by differential position exchange are higher for N on ant,v = 32 with respect to N on ant,v = 8. This difference is explained considering that the pointing error has a stronger impact on the communication performance when a narrower BW is employed. Focusing on Fig. 6a, the maximum distance between the curves for N pos bit = 6 and N pos bit = 10 is in the order of 1 dB and the N pos bit = 10 case is superimposed to the one employing absolute position. In Fig. 6b, on the other hand, the gap between N pos bit = 6 and N pos bit = 10 increases and the latter is no longer superimposed to the absolute position case, although the gap is almost negligible (0.1 dB). We can conclude that, as the BW reduces (e.g., for N on ant,v = 32), the performance is more sensitive to the quantization errors and a higher signalling overhead is required.

C. IMPACT OF COVARIANCE QUANTIZATION
For the analysis on the covariance signalling, we compare the performances of the different strategies described in Section III-B: the full covariance signalling, the differential one and the scalar covariance. For the following analyses, we employ the differential strategy for position encoding. Observing that, introducing BW adaptation, the SA architecture rarely relies on the full MIMO hardware, we select N pos bit = 6, which is a good trade-off to limit the overhead and the quantization errors. Further details concerning this aspect are given in the next paragraphs. Fig. 7 shows the SNR loss of two fixed BW methods, with N on ant = 8 and N on ant = 32 respectively, and three adaptive BW methods, employing full, differential and scalar covariance signalling. The position signalling is performed with the differential approach and N pos bit = 6, both for the fixed and adaptive BW methods. We first observe that the additional losses introduced by the differential signalling of the covariance are lower than 1 dB. This is a promising result since the required signalling for the covariance information is halved with respect to the full covariance approach. Secondly, the results highlight the improvements brought by the BW optimization, which, thanks to the covariance signalling, ensures better communication performances with respect to the fixed BW case. The final remark concerns the scalar covariance signalling. The dashed red curve is almost superimposed to the one of the differential covariance, hence no significant differences in terms of SNR loss can be noticed between the two methods. This confirms that the terms that are neglected when exchanging the scalar covariance are not relevant for the proposed BW adaptation.
To deepen the analysis, in Fig. 8 we report the distribution of the number of selected active antennas differential and scalar covariance signalling methods. We notice that scalar covariance is more conservative with respect to the full covariance one, since it tends to select a smaller number of active antennas. This behaviour is explained considering that . When σ XX,v (t) and σ YY,v (t) are encoded with consecutive variance levels of Table 1 (e.g., 0.5 m and 1 m), the uncertainty region of the scalar covariance approach is a circle whose radius coincides with the major semi-axis of the original ellipse. The area where vehicle v is expected to lie leads to the selection of a wider beam (i.e., less active antennas). However, we would like to remark that the reduced number of active antennas does not imply a lower V2V capacity. To prove this, in Fig. 8b we plot the mean capacity versus the number of active antennas. We notice that, overall, the capacity is pretty stable around 6.5 Gbps. It follows that BW adaptation ensures a stable V2V link, which automatically adapts the BW to the uncertainty of antenna information and V2V distance. As a further benchmark, we remark that the average capacity for the fixed BW methods with 8 and 32 active antennas is of 6.5 Gbps and 5.6 Gbps, respectively. This suggests that the current accuracy of vehicle positioning hinders the potential of the MIMO system with a high number of fixed antennas (i.e., narrow beams), as pointing errors introduce significant losses on the MIMO gain.
Besides a reduction of the SNR loss, the adaptive methods ensure better performances also in terms of stability, evaluated with the fading duration. Fig. 9 shows the ECDFs of the fading duration for the considered SA methods and the CAM-based BAT one. The latter is confirmed to be unable to guarantee reliable V2V communication, especially for narrow beams. For N on ant = 32, the duration of fading when CAM signalling is used is above 10 ms in 40% of the cases. Adaptive methods guarantee a fading below 3 ms in 95% of the cases, whereas the fixed BW ones do not provide such good results, with a fading duration larger than 10 ms in 95% of the cases, which does not meet the requirements of autonomous driving [4].
A final comparison between fixed and adaptive BW methods regards the channel capacity. Fig. 10 reports the ECDF of channel capacity, highlighting that the use of a fixed BW with N on ant,v = 8 guarantees higher stability of the channel, with a capacity always larger than 5 Gbps but bounded to a maximum of 8 Gbps. On the other hand, with N on ant,v = 32, a higher antenna gain is experienced and a larger maximum capacity (9 Gbps) is ensured but, at the same time, it has an extremely high sensitivity to pointing error that makes this option unsuitable to satisfy the V2V communication constraints [4]. Adaptive BW methods combine link stability and capacity, by adjusting dynamically the BW. These results on the impact of covariance quantization on the communication performance suggest that, rather than devoting a large number of bits for position signalling, it is advisable to enable BW optimization through the dissemination of covariance information, especially considering that the additional overhead of the proposed method is in the order of decimals of kbps.
The results included in this section are consistent with the state-of-the art ones. The SNR loss shown in Fig. 6 is comparable to the one included in [45], where the misalignment causes a loss below 3 dB in 95% of the cases. The same statistics is valid for the SA methods proposed in this work, provided that N pos bit ≥ 6. Similarly to the results in [67], the designed BW adaptation is capable of managing extremely narrow beams (N on ant,v (t) = 32, φ v = 3.2 deg), but without the computational burden of an optimization algorithm. Focusing on the link capacity, the values achieved by the proposed BAT methods are compatible with the findings in [46], provided that the bandwidth is scaled (we used a B = 400 MHz instead of 2.16 GHz).

V. CONCLUSION
This paper proposed a number of signalling strategies to support the introduction of SA beam pointing control for mmWave V2V communications. The considered cooperative BAT architecture requires V2V sharing of the estimate of antenna position and its covariance, to track the LOS direction and adapt the dimension of the beam according to the quality of the pointing information. The proposed strategies are evaluated on a realistic simulated urban scenario. The results show that, by exploiting differential approaches for both the position and covariance encoding, the V2V communication system achieves performances compliant with CAV requirements, with moderate signalling overhead. Furthermore we highlighted the importance of adopting a reconfigurable adaptive hardware to cope with the time varying uncertainty of the link estimate. The BW adaptation, indeed, provides a more stable communication with respect to the fixed BW case, at the cost of few kbps of additional signalling.
Future developments will concern the extension of the number of on-board sensors to enhance the accuracy of the pose estimation and better exploit the capabilities of the MIMO system, by employing smaller BW dimensions for a longer time. Moreover, we will consider a more complex scenario, modelling a larger vehicular network, where it is mandatory to model the effects of multipath propagation, interference and inter-vehicular blockage.