Abstract： Estimating time?selective millimeter wave wireless channels and then deriving the optimum beam alignment for directional antennas is a challenging task. To solve this problem， one can focus on tracking the strongest multipath components （MPCs）. Aligning antenna beams with the tracked MPCs increases the channel coherence time by several orders of magnitude. This contribution suggests tracking the MPCs geometrically. The derived geometric tracker is based on algorithms known as Doppler bearing tracking. A recent work on geometric?polar tracking is reformulated into an efficient recursive version. If the relative position of the MPCs is known， all other sensors on board a vehicle， e.g.， lidar， radar， and camera， will perform active learning based on their own observed data. By learning the relationship between sensor data and MPCs， onboard sensors can participate in channel tracking. Joint tracking of many integrated sensors will increase the reliability of MPC tracking.
　　Keywords： adaptive filters; autonomous vehicles; directive antennas; doppler measurement; intelligent vehicles; machine learning; millimeter wave communication
　　1 Introduction
　　illimeter wave （mmWave） frequency bands have been a candidate for vehicular communication for several decades [1]-[3]. MmWave train?to?infrastructure path loss was measured in [2]， while the transmission behaviour of mmWave for communication between vehicles was examined in [1]. Recent advances in mmWave circuit technology have aroused interest in mmWave vehicular communication [3] and in joint vehicular communication and radar [4]. MmWaves offer large bandwidths and enable raw data exchange between vehicles [5]. The main problems with vehicular mmWave communication are the direct proportionality of the maximum Doppler shift and the carrier frequency as well as the beam alignment challenge in the dynamic environment. In [6] and [7]， however， it has been shown theoretically that directional antennas intended for mmWaves function as spatial filters. The Doppler effect and thus the time selectivity is drastically reduced by beamforming. This is shown experimentally in [8] and [9]. There seems to be a consensus that channel tracking tackles the second challenge of the dynamic environment [10]-[21]. Channel tracking is the process of causally estimating the current or future direction of the line?of?sight （LOS） component or other strong multipath components （MPCs） based on previous measurements. The main advantage of channel tracking is the extended coherence time after successful beamforming. The channel coherence time of the beam aligned channel is several orders of magnitude longer than that for omnidirectional reception [7]. A subsequent channel estimation therefore runs on a coarser time grid. 　　The work in [10] adopts the idea and formalism of [21] and applies them directly to THz lens antennas. Extended Kalman filters are used in [11]， [18]， and [19] to track the beam directions based on channel gain measurements. In [15]， domain knowledge is used and it is argued that the road implicitly determines the direction in which a vehicle is expected. Beam training is avoided by using this geometric prior knowledge. Assuming a constant angular acceleration that is motion along circles， [20] proposes an algorithm based on the unscented Kalman filter. Probabilistic beam tracking is suggested in [16]. Moreover， in [13] and [14] the stochastic Newton method is used， and these algorithms surpass IEEE 802.11ad based approaches and compressive sensing based approaches [17]; the work in [13] and [14]， shows good performance for angular velocities of up to 5°/s.
　　In [22]， it was first proposed to utilize the Doppler information for mmWave beam tracking. Measurements in [23] clearly demonstrate that interacting objects， such as overtaking cars， produce distinguishable MPCs in the Doppler profile. The proposed algorithm herein， exploiting Doppler information， is assessed in scenarios where the angular velocity exceeds 100°/s for a short duration.
　　This contribution proposes to track the MPCs geometrically given quantized angular （azimuth） measurements and noisy Doppler observations. The quantized angular information is obtained by an analog or hybrid beamforming array or a dielectric lense [24]. “Geometric” refers to the [（x，y）] coordinates originating in the antenna array and the relative velocity [（x，y）] to the receiver motion. We assume that the transmitter， the receiver， and the interacting objects move without acceleration. Under these assumptions， algorithms performing target?motion analysis by means of Doppler?bearing measurements [25]-[27] are directly applicable. The work in [25]-[28] proposes a formulation called “pseudolinear.” Pseudolinear refers to a formulation where the nonlinearities are either hidden in a measurement （regression） matrix or are lumped within the noise term. This leads to the undesirable consequence of noise correlation of the measurements and the measurement matrix， eventually leading to biased solutions [26]. An early work [25] removes this bias by the method of instrumental variables. Due to a potential divergence of the instrumental variables approach [27]， later work [26]， [27] employs the method of total least squares. To apply total least squares， error covariance matrices must be known . The proposed approach is inspired by [27]， but does not need knowledge about the error covariances. 　　In addition to the excellent angular tracking performance of the proposed algorithm， the obtained geometric information of the MPCs can be utilized to learn the MPCs from other sensors on board of automated vehicles [5]. The concept of using external information for improved channel estimation was recently re?introduced， see [29] and [30] and the reference therein. Machine learning for configuring wireless links has also been proposed in the context of WLAN and mobile communications [31]-[33]. The actual machine learning implementation is not within the scope of this contribution. This contribution focuses on a framework for active learning of beam alignment. This paper is an extended version of [34].
　　（1） Brief Review of Geometric Tracking or Wireless Positioning：
　　Ground?based radio?frequency localization has become an established technique. Based on known anchor positions， techniques such as fingerprinting， hop counts， receive signal strength， time?（difference）?of?arrival， frequency?difference?of?arrival， and angle?of?arrival are at hand [35]. Knowing the position of the communication partner is extremely valuable for the task of beam alignment [36]. In [37]， a mmWave base station was equipped with a 360° camera; both positional information sources—vision and the mmWave link—were fused to enhance the precision. The situation changes however once vehicle?to?vehicle communication is considered. There， mainly GNSS positions of communication partners are exchanged by low?rate messages [38]. Future automated self?driving cars will be equipped with a plurality of sensors and will thereby perform massive sensing [5]. The smart use of all of these sensors will renders it possible to determine the position of communication partners solely by onboard sensors.
　　（2） Notation：
　　Matrices [Z] and vectors [z] are denoted by bold letters. The all zeros vector （matrix） is expressed by [0] and the identity matrix is expressed by [I]. The Euclidean norm is symbolized by [?]. A quantity defined with a start index [i] and stop index [k] is indicated via the subscript [（ ? ）i：k]. Estimated quantities are marked with （ ^. ） . The four?quadrant inverse tangent is denoted by [arctan?，?]. The dagger [（ ? ）?] is used for pseudo inverses and [（ ? ）T] is used for transposition.
　　2 Active Learning by Onboard Sensors
　　The idea behind active learning is to actively select the “optimal” training data. For some applications statistically optimal choices are computable [39]. Selective sampling [40] is a rudimentary form of active learning and especially suited for problems where the cost of labelling is high. The survey paper [41] provides a good introduction to active learning. According [41]： “The key idea behind active learning is that a machine learning algorithm can achieve greater accuracy with fewer labelled training instances if it is allowed to choose the data from which is learns.” 　　On board of automated （self?driving） cars， there will be sensors such as global navigation satellite systems （GNSSs）， automotive radars （for automatic cruise control and collision detections）， lidar （for measuring distances to other objects）， and 360° camera vision systems. All these sensors have in common that they track objects via target states [42]. At the simplest， this target state consists of the relative [（x，y）] position and the relative velocities [（x，y）].
　　Due to the high?resolution of lidar， radar， and vision， self?driving cars produce several gigabytes of data per second and hence hundreds of terabytes per day [43]， [44]. Processing all these data for the indented use case of driving poses already a challenge and more and more tasks are already shifted towards fog and cloud computing units [45]， [46]. To use these gigabytes of data for the tracking of MPCs， every tracked object of the onboard sensors must be labelled as “MPC” or “no MPC” （pedestrian， non?communicating car， static objects， etc.）. This leads to high labelling efforts and to a huge amount of training data where most of labels will be “no MPC”.
　　The key idea is now to exploit the geometric position of the MPC and thus to only label those targets that are in the vicinity of the MPC. The process of associating MPCs to “targets” is illustrated with black circles in Fig. 1. Instead of human （or any other oracle） labelling there is an active choice of the system which targets to consider for learning the beam alignment. After a successful learning phase， all sensors on board should later do the channel tracking. By using machine learning， one can eliminate or significantly reduce the beam measurements needed for the currently proposed tracking algorithm. If for all of these target states it is known whether they belong to the LOS component or to a specular reflection， the onboard sensors will track the MPCs.
　　In this sense this paper provides an algorithm which determines the geometric positions of the communication partners in order to label them as interesting training samples.
　　3 Measurement， Regression， and Projection
　　Model
　　The regression model is based on the model proposed in [27]. The main idea of [27] is to track non?accelerating objects on linear trajectories in polar coordinates; target motion analysis in polar coordinates yields a smaller bias than in rectangular coordinates. The regression model is hence formulated in polar coordinates. This idea is illustrated in Fig. 2. The original tracking problem of [27] uses a running reference （blue）. Thereby at each time the current state is estimated. This approach produces an increasing system of equations， anew， at any time. In contrast to [27]， the proposed algorithm will use a fixed reference （red） and gather only one new equation per time step. Thereby the estimate of the initial state is refined and its accuracy is improved over time， as in [25]. Through this reformulation， the initial state?vector is estimated recursively. The state vector at current and future times is predicted by a projection. 　　3.1 Quantized Angular Measurements by the UCA
　　 Codebook
　　For target motion analysis a noisy bearing （angular） observation is assumed where the noise is usually modelled Gaussian [27]. In this study， however， quantized angular observations will occur. A 60 GHz uniform circular array （UCA） with [N=64] elements equidistantly spaced on a radius of [rUCA=N/2?λ/2≈8] cm is used. The half power beam width is [θ3dB≈2π/N≈6?]. The UCA is inherently symmetric in its azimuthal resolution. In contrast to uniform planar arrays， the UCA beam pattern does not change with the pointing direction. To save cost， analog precoding （beamforming） with 4 bit RF phase?shifters is employed. The phase shifts are pre?computed in a codebook spaced by [θ3dB2] which gives [2π2πN2] [=2?N=128] codebook entries. Beampattern of the UCA are shown in Fig. 3.
　　3.2 Regression Model
　　6 Simulations
　　The simulations focus on line?of?sight scenarios. Note that the proposed approach works for specular reflections as well （see next section）. Clustered reflections will lead to a higher uncertainty in determining the azimuth angle. Similar to the IEEE 802.11ad standard， it is assumed that the TX is transmitting its reference signal omni?directionally. Initially， the RX is scanning all entries from the codebook and determines the direction towards the TX. We compare the performance if this procedure is repeated every 20 ms or 50 ms. After the 10th iteration， the projection （11） is used to predict the future azimuth angle. Having the projected azimuth angles at hand， the algorithm only probes the closest three codebook entries for 20 ms update rate or five codebook entries for 50 ms update rate. This gives a speed up of a factor [128/3≈43] or [128/5≈26] as compared to a full codebook scan.
　　The first scenario， entitled “half?overtaking”， starts when the overtaking， red， TX car is at the same height as the slower， black， RX car. The overtaking car has 20 m/s excess speed and is observed for 3 s. In the second scenario， entitled “full?overtaking”， the TX starts behind the RX and overtakes with an excess speed of 10 m/s. The manoeuvre is now observed for 6 s. The lateral distance was chosen such that the resulting maximum angular velocity [ωmax=v/rmin=（20  m/s）（8  m）=][（10  m/s）（4  m）=2.5 rad/s≈140?/s]is equal in both scenarios. The presented Monte Carlo mean is calculated from 10 000 runs. To obtain different channel realizations， the lateral distance is varied uniformly in [Δrmin?U（-1  m，1 m）] around the mean lateral distances， and the angle between both cars is varied uniformly in [φ?U-2?，2?]. These variations are drawn within the sketch of the manoeuvre as black arrows in Fig. 5a. The normalized mean squared error of the prior work [27]， the proposed sequential implementation from Section 4， and both error bounds from Section 5 are plotted in Figs. 5c （20 ms update rate） and 5e （50 ms update rate）. Figs. 5b and 5d show the respective scatter plots of the estimated [x，y] position of the proposed sequential estimator for the first Monte Carlo run. 　　Half overtaking （with update rate of 20 ms） turns out to be not so burdensome than full?overtaking. That is because right from the beginning， the TX car is seen at different azimuth angles and close to the initial solution of （0，0） and the algorithm convergences fast. The regression model of [27] suffers from a strong bias due to the error correlation of the current observation and the regression matrix. The sequential algorithm outperforms the prior non?sequential modelling approach. The “errors?in?variables” approach comes very close to the error?free regression matrix. Furthermore， there is only a small loss to the nullspace projection. Keep in mind that the “error?free regressors” and the nullspace projection approach make use of exact （yet unknown）， unquantized azimuth angles！ The full overtaking manoeuvre is characterized by a difficult geometry. At the beginning the TX car is always seen at the same codebook index and the algorithm struggles to converge. In this region the         algorithm is used to prevent divergence. After approximately 0.5 s， three different azimuth angles have been measured and the algorithm hands over to RLS. Even with an estimate of the initial state and the covariance matrix， the RLS algorithm needs a considerable time to converge afterwards. The situation is aggravated by the fact that the toughest part （TX car closest to RX car [→ωmax]） comes before convergence sets in. Nevertheless， an acceptable tracking result can be achieved here as well.
　　For an update rate of 50 ms the peformance loss at “full?overtaking” is minor. In contrast， the previously simpler case of “half?overtaking” has now a larger performance loss. Due to the slower update rate， after only a few measurements， the overtaking car is already at steeper angles where the Doppler shift does not change so much and convergence is harder to achieve.
　　7 The LOS Blocked Scenario
　　Until now， all cases considered LOS. The proposed tracker， however， is also applicable to scenarios with specular reflections. Fig. 4 shows a blockage scenario. The direct LOS between TX （the red car） and RX （the black car） is blocked by the green car. If another car overtakes this platoon， it can act as reflector and can be tracked by its Doppler shift. The feasibility of this approach has been verified experimentally in [9] and [23]. The Doppler shifts of overtaking vehicles produce very distinct Doppler traces. Such an exemplary trace is illustrated on the right?hand side of Fig. 4. The only adaptation for the algorithm is a factor 2 occuring in the Doppler shift equations. 　　8 Conclusions
　　Geometric tracking of specular multipath components in vehicular millimeter wave channels is possible with low complexity algorithms. The proposed algorithm achieves good tracking even under very dynamic scenarios. This considerably relaxes the time required for beam training. In addition， the proposed algorithm outputs a state vector that reflects the relative position and velocity of the multipath components. With this knowledge， it is possible to label the targets for onboard sensors as multipath components. This enables active learning for onboard sensors.
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