Radar-Assisted Multiple Base Station Cooperative mmWave Beam Tracking

Abstract

In the future vehicular networks with an increased number of transceiver antennas and higher vehicle speeds, more frequent beam switching is required to ensure the quality of communication, which poses challenges to beam tracking speed and resource efficiency. Integrated sensing and communication (ISAC) provide a new solution to cope with this problem since radar echo can help to predict the vehicle’s future location and beam direction. Therefore, we present a radar-assisted beam tracking algorithm based on Extended Kalman filtering (EKF) and multi-road side unit (RSU) cooperation in this article. Each RSU uses EKF and radar echo to predict and track the vehicle position and upload the prediction information to the edge server (ES). By deploying multiple RSUs, the ES uses the uploaded distributed sensing information for joint estimation and thus improves the accuracy of vehicle location prediction, which is used for the beam tracking task at the next moment. Considering the real complex road conditions, we investigate two scenarios where vehicles move linearly or curvilinearly. Simulation results show that the proposed method with multiple base station cooperation improves the spectral efficiency by 34% and 20% over non-cooperative beam tracking in linear and curvilinear mobility, respectively. In addition, compared with traditional beam tracking based on beam scanning and signaling feedback, radar-assisted beam tracking significantly reduces the communication overhead.

1. Introduction

Millimeter-Wave (mmWave) frequency band has attracted significant attention due to its abundant spectrum resources and high data transmission rate [1]. Although the path loss of mmWave is very high during propagation [2], the directional gain provided by beamforming can compensate for this disadvantage. Therefore, the performance of millimeter wave systems is closely related to the accuracy and timeliness of beam management. Conventionally, during the initial access, beam training periodically transmits and receives pilots at all the possible beams to search for the optimal beam pair. Furthermore, beam tracking uses only a small number of pilots to update the beam information by tracking the change of channel state information (CSI) over time. During the mobility of the vehicle, the CSI of adjacent moments is time-correlated. As a result, the CSI at the previous moment is used as prior information, and the channel time correlation at adjacent moments can be used to predict the CSI at the current moment. Beam switching then takes place when necessary in order to keep track of the moving vehicle.

In high mobility scenario, the fast-moving vehicle can cause rapid channel variations, short channel coherence time and thus more frequent beam switching. Meanwhile, the large-scale antenna configuration leads to an increased number of beams on transceivers and narrower beam width. In this case, conventional beam tracking schemes will inevitably incur considerable latency and communication overhead. Therefore, how to reduce the overhead of beam tracking and perform efficient beam switching has become a significant challenge in the high mobility VEHICLE-TO-EVERYTHING (V2X) scenario.

Conventional solutions for beam tracking can be divided into two types. The first type is to reduce CSI dimensionality by tracking some parameters of the channel. As the direction of the optimal beam is closely related to the angle of the physical channel, many works optimize beam switching by tracking the angular parameter of the channel, i.e., Angle of Arrival (AOA) or Angle of Departure (AOD). In [3], an EKF-based AOA/AOD tracking algorithm was proposed. In [4], the authors investigated the problem of beam tracking for UAV scenarios and proposed to predict the AOA and AOD in a 3D channel based on an Unscented Kalman Filter (UKF). The authors in [5,6] proposed to transform the spatial domain CSI into the angular domain CSI by DFT and use linear Kalman filtering (LKF) to track the angular domain CSI in a high-speed railway scenario and then complete beam switching. In [7], an EKF-based high-speed railway terminal location prediction algorithm was proposed. Pioneered by [7], authors in [8] proposed a novel nonlinear state transfer model, in which the position, speed, forward direction and dynamically correlated angular velocity variance of the train are jointly considered to further improve the position prediction and beam switching accuracy. However, the above methods still require some signaling feedback and interaction at each moment, as well as the scenario is relatively simple and idealized. The second method for beam tracking uses neural networks to predict CSI. In [9], the authors proposed to predict narrowband channel CSI using Full Connected Recurrent Neural Network (FCRNN). In [10,11], AOA/AOD, channel parameters, power, and distance are used to construct a beamforming fingerprint database for beam switching based on deep learning techniques. Some work focused on the design of neural networks suitable for communication. For example, a learning framework combining Convolutional Neural Networks (CNN) and Long Short-Term Memory (LSTM) for predicting CSI was designed in [12]. However, the performance of the second method is significantly affected by the source of training data and the structure of neural networks. Only in the scenarios where enough training data is acquired can the methods have great performance.

It’s noteworthy that the emerging of Integrated sensing and communication (ISAC) technology provides a new effective solution to beam tracking in high-speed vehicle mobility scenarios. As the typical application of ISAC, dual-function radar communication (DFRC) system uses joint hardware equipment to provide dual functionalities, i.e., radar sensing and communication. Thus, it’s possible to use the function of radar sensing to assist beam tracking in communication. For example, a novel transceiver architecture and frame structure for a DFRC system operating in the mmWave band were proposed in [13]. Yet this scheme is not tailored for vehicle mobility scenarios. Further, the authors in [14] extracted significant information from radar signals to perform beam alignment for the mmWave vehicle-to-infrastructure (V2I) scenario. However, this scheme separates the communication system from the radar system, an extra radar device will inevitably result in high hardware costs. In [15], a millimeter-wave joint radar-communications (JRC) system comprising a bi-static automotive radar and vehicle-to-vehicle (V2V) communications was proposed, but the high mobility of the vehicle is not addressed. The authors of [16] proposed a radar-assisted EKF scheme to perform beam tracking by predicting the kinematic parameters of the vehicle in the mmWave band. Furthermore, based on [16,17] exploited Bayesian framework-based message passing techniques for predicting vehicle mobility parameters, which can reduce computational complexity. However, the above two schemes only model linear mobility scenarios, which are difficult to meet the requirements of complex curvilinear mobility scenarios. In addition, the authors in [18] proposed to explore the use of intelligent reflecting surfaces (IRS) in OFDM-DFRC systems to enhance sensing and communication performance by taking advantage of the passive beamforming gain.

To further enhance the tracking precision of DFRC systems under high mobility scenarios, we propose a multi-RSU cooperative mmWave beam tracking algorithm based on radar echo and EKF prediction in this paper. Our scheme not only removes the feedback and pilot signaling overhead in conventional solutions but also improves the prediction accuracy of the kinematic parameters and beam tracking performance by multi-RSU cooperation. Considering the real complex road conditions, we studied both linear and curvilinear mobility scenarios and derived the vehicle mobility model and EKF iteration process for different trajectories. More specifically, we employ a downlink DFRC system, where the DFRC signal completes both sensing and communication tasks. The signal transmitted by the RSU via each beam is partially reflected the transmitter by the body of the vehicle and is also partially received by the vehicle’s antenna array. The radar echoes are used for angle, distance and speed prediction after matched filtering and the EKF, while the signal received by the vehicle is used for communication. Each RSU uploads the predicted kinematic parameters to the ES, where the optimal vehicle predicted position is calculated based on the weights. Following this, the ES downlinks the optimal location prediction to each RSU as the input of the EKF’s next time slot prediction. The main contributions of our work are summarized as follows:

• We propose a radar-assisted and multi-RSU cooperation mmWave beam tracking algorithm based on joint sensing and communication for V2I networks. The proposed method does not require downlink pilot and uplink feedback and significantly reduces latency and communication overhead. In contrast with non-cooperative beam tracking, multi-RSU cooperation embodies the advantages of distributed sensing, which can provide higher prediction accuracy and more efficient beam tracking.
• We investigate the prediction of the vehicle’s position in two mobility scenarios, i.e., linear and curvilinear, respectively. The kinematic model, EKF state evolution model and the solution of the Jacobi matrix are firstly derived for radar-assisted beam tracking in curvilinear mobility scenarios.
• We propose to use the more realistic extended Saleh-Valenzuela (eSV) channel model in the DFRC system and consider the delay between different paths to expand the narrowband channel to the wideband channel.

2. System Model

We consider a radar-assisted mmWave V2I network as depicted in Figure 1, where each RSU serves K high mobility vehicles. Each RSU has deployed a downlink DFRC system which is equipped with a mmWave massive MIMO uniform linear array (ULA) consisting of $N_t$ transmit antennas and $N_r$ receive antennas, while each vehicle is also equipped with a ULA array consisting of M antennas. As the computing center of multi-RSU cooperation, ES can receive reports from RSUs and issue cooperation results to each RSU. At a certain moment, a vehicle can only be served by one RSU, while other RSUs only track vehicle movement trajectory by radar echoes. For simplicity, we assume that the Line-of-Sight (LoS) path always exists between RSU and the vehicle, and the radar echoes are reflected through the LoS path. Figure 1 shows the linear mobility scenario where the vehicle trajectory is parallel to the antenna array of the RSU, thus the AOD for beamforming and the AOA for beam combining are equal. For the curvilinear mobility scenario, the case is different and will be illustrated later.

To establish dependable connections for communication, each RSU needs to acquire the accurate azimuth angle of the vehicles, and similarly, each vehicle also needs to acquire the RSU’s relative angle. After acquiring angle information, the RSU and the vehicle could both utilize their antenna arrays to create narrow beams, enabling precise alignment toward each other. During initial access, a conventional beam searching scheme could be employed to acquire initial estimation, such as angle, distance, speed, etc. For the linear mobility scenario, prediction is divided into two steps. In step I, the estimate of the AOD at the ($n-1$)-th slot, denoted as ${\widehat{\theta }}_{n-1}$, is used to predict the AOD at the n-th slot, denoted as ${\widehat{\theta }}_{n|n-1}$. In step II, ${\widehat{\theta }}_{n|n-1}$ is used for predicting AOD at the ($n+1$)-th slot, denoted as ${\widehat{\theta }}_{n+1|n-1}$. At time slot n, the RSU formulates directional transmit beams aimed at vehicles by using ${\widehat{\theta }}_{n|n-1}$, and the DFRC signal transmitted by RSU contains the step II prediction ${\widehat{\theta }}_{n+1|n-1}$, which is used to formulate receive beams at the vehicle at the $(n+1)$-th slot. It should be noted that ${\widehat{\theta }}_{n+1|n-1}$ is not suitable for beam combining on the vehicle side for curvilinear mobility scenarios since AOD is not equal to AOA. Moreover, the partial DFRC signal reflected by the vehicle is used to calibrate predicted parameters at time slot n. The kinematic parameters that have been calibrated are subsequently utilized as inputs for the Extended Kalman Filter (EKF) in the $(n+1)$-th and $(n+2)$-th slots. In what follows, we will introduce the vehicle mobility model and signal processing model.

2.1. Vehicle Mobility Evolution Model

In order to use the EKF to accurately predict the angles needed for beamforming, a precise model of vehicle kinematic evolution is particularly necessary. Many existing methods focus on linear kinematics, as depicted in Figure 2a, while we propose a novel vehicle curvilinear kinematics model based on previous research, as shown in Figure 2b.

For the linear mobility scenario, at the n-th slot, $d_n$ denotes the distance between RSU and the vehicle, ${\theta }_n$ denotes the angle of the vehicle relative to the RSU, $v_n$ denotes the speed of the vehicle, $\Delta d$ denotes the distance moved by the vehicle in a straight line at adjacent moments, $\Delta \theta$ denotes the variation of the angle at adjacent moments, ${\beta }_n$ denotes the reflection coefficient which contains both the radar cross-section (RCS) of the vehicle and the path-loss of the signal, specifically, ${\beta }_n=\frac{{\epsilon }_n}{2d_n}$, where ${\epsilon }_n$ denotes fading coefficient based on the radar cross-section. Both transmit and receive beams are formulated based on the angles predicted by step I and step II as mentioned previously. According to [16], the linear mobility evolution model is as follows,

$$\left\{\begin{array}{c}{\theta }_n={\theta }_{n-1}+d_{n-1}^{-1}{\nu }_{n-1}\Delta Tsin{\theta }_{n-1}+q_{\theta }\\ d_n=d_{n-1}-{\nu }_{n-1}\Delta Tcos{\theta }_{n-1}+q_d\\ {\nu }_n={\nu }_{n-1}+q_{\nu }\\ {\beta }_n={\beta }_{n-1}+\left(1+d_{n-1}^{-1}{\nu }_{n-1}\Delta Tcos{\theta }_{n-1}\right)+q_{\beta }\end{array}\right.$$

(1)

where $\Delta T$ denotes the time interval between adjacent slots, $q_{\theta }$, $q_d$, $q_v$, $q_{\beta }$ denote Gaussian noise which has a mean of zero and a variance of ${\sigma }_{\theta }^2$, ${\sigma }_d^2$, ${\sigma }_v^2$, ${\sigma }_{\beta }^2$, respectively. For the curvilinear mobility scenario, we consider the curvilinear mobility of the vehicle as the composition of two uniformly accelerated linear moving objects. It is worth noting that, to guarantee that the trajectory of the vehicle is curvilinear, the direction of the combined velocity and the direction of the combined acceleration at the initial moment could not be the same. Therefore, based on kinematic knowledge and the illustration in Figure 2b, we can obtain the following geometric and kinematic equations.

$$\left\{\begin{array}{c}\Delta x=v_{x_{n-1}}\Delta T+\frac{1}{2}{a}_x\Delta T^2\\ \Delta y=v_{y_{n-1}}\Delta T+\frac{1}{2}{a}_y\Delta T^2\\ \Delta d=\sqrt{\Delta x^2+\Delta y^2}\\ \phi =arctan\left(\frac{\Delta y}{\Delta x}\right)\\ {d_n}^2={d_{n-1}}^2+\Delta d^2-2d_{n-1}\Delta dcos\left({\theta }_{n-1}-\phi \right)\\ \frac{\Delta d}{sin\Delta \theta }=\frac{d_n}{sin\left({\theta }_{n-1}-\phi \right)}\end{array}\right.$$

(2)

where $\Delta x$ and $\Delta y$ denote the distance that the vehicle moves in the horizontal (X-axis) and vertical direction (Y-axis), respectively. $v_{x_{n-1}}$ and $v_{y_{n-1}}$ denote the velocities of the vehicle in the X-axis and the Y-axis at the $(n-1)$-th slot, respectively. $a_x$ and $a_y$ denote the acceleration of the vehicle in the X-axis and Y-axis, respectively, which are constant during the vehicle’s mobility. $\phi$ denotes an angle used for calculation. Similar to the linear mobility scenario in [16], the curvilinear mobility evolution model is as below,

$$\left\{\begin{array}{c}{\theta }_n={\theta }_{n-1}+\Delta \theta +q_{\theta }\\ d_n=d_{n-1}-\Delta dcos({\theta }_{n-1}-\phi )+q_d\\ v_{y_n}=v_{y_{n-1}}+a_y\Delta T+q_{v_y}\\ v_{x_n}=v_{x_{n-1}}+a_x\Delta T+q_{v_x}\\ {\beta }_n={\beta }_{n-1}\left(1+\frac{\Delta dcos\left({\theta }_{n-1}-\phi \right)}{d_{n-1}}\right)+q_{\beta }\end{array}\right.$$

(3)

where $q_{\theta }$, $q_d$, $q_{v_y}$, $q_{v_x}$, $q_{\beta }$ denote Gaussian noise which has a mean of zero and a variance of ${\sigma }_{\theta }^2$, ${\sigma }_d^2$, ${\sigma }_{v_y}^2$, ${\sigma }_{v_x}^2$, ${\sigma }_{\beta }^2$, respectively.

2.2. Radar Signal Model

The DFRC signal transmitted via $N_t$ antennas at the RSU can be expressed as

$${\tilde{\mathbf{s}}}_n\left(t\right){=\mathbf{F}}_n{\mathbf{s}}_n\left(t\right)\in {\mathbb{C}}^{N_t\times 1}$$

(4)

where ${\mathbf{s}}_n\left(t\right)={\left[s_{1,n}\left(t\right),...,s_{K,n}\left(t\right)\right]}^T\in {\mathbb{C}}^{K\times 1}$ denotes the transmitted K downlink DFRC streams corresponding to K vehicles at the n-th slot, ${\mathbf{F}}_n=\left[{\mathbf{f}}_{1,n},{\mathbf{f}}_{2,n}...,{\mathbf{f}}_{K,n}\right]\in {\mathbb{C}}^{N_t\times K}$ denotes transmit beamforming matrix, where each column represents a vehicle’s beamforming vector. Therefore, the reflected signal echoes received at the RSU by vehicles can be formulated as [16]

$$\begin{array}{c}{r}_n\left(t\right)=\kappa {\displaystyle \sum _{k=1}^K}\sqrt{p_{k,n}}{\beta }_{k,n}{e}^{j2\pi {\mu }_{k,n}t}\mathbf{v}\left({\theta }_{k,n}\right){\mathbf{u}}^H\left({\theta }_{k,n}\right){\tilde{\mathbf{s}}}_n\left(t-{\tau }_{k,n}\right)+{\mathbf{w}}_r\left(t\right)\end{array}$$

(5)

where $\kappa =\sqrt{N_t{N}_r}$ is the antenna array gain, $p_{k,n}$ is the transmit power for the n-th slot to the k-th vehicle, ${\tau }_{k,n}$, ${\mu }_{k,n}$, ${\beta }_{k,n}$ denote the time-delay, the Doppler frequency and the reflection coefficient for the k-th vehicle, respectively. ${\mathbf{w}}_r\in {\mathbb{C}}^{N_r\times 1}$ denotes Gaussian noise with zero mean and variance of ${\sigma }^2$. $\mathbf{u}\left(\theta \right)$ and $\mathbf{v}\left(\theta \right)$ are transmit and receive antenna array responses, which yield

$$\begin{array}{c}\mathbf{u}\left(\theta \right)=\sqrt{\frac{1}{N_t}}{\left[1,e^{-j\pi cos\theta },...,e^{-j\pi \left(N_t-1\right)cos\theta }\right]}^T\end{array}$$

(6)

$$\begin{array}{c}\mathbf{v}\left(\theta \right)=\sqrt{\frac{1}{N_r}}{\left[1,e^{-j\pi cos\theta },...,e^{-j\pi \left(N_r-1\right)cos\theta }\right]}^T\end{array}$$

(7)

where we make the assumption that the antenna spacing of the ULA is half-wavelength. The beamforming vector is formulated based on the predicted angle, specifically, the k-th column ${\mathbf{f}}_{k,n}$ of ${\mathbf{F}}_n$ is:

$$\begin{array}{c}{\mathbf{f}}_{\mathrm{k},\mathrm{n}}=\mathbf{u}\left({\widehat{\theta }}_{\mathrm{k},\mathrm{n}|\mathrm{n}-1}\right),k=1,2,...,K\end{array}$$

(8)

However, although RSU will receive echoes from multiple vehicles, the inter-beam interference can be neglected given the antenna array response are asymptotically orthogonal to each other under the massive MIMO system [19,20]. Therefore, there is no interference between reflected echoes from different vehicles. The radar echo received by the RSU for the k-th vehicle at n-th slot is expressed as

$${\mathbf{r}}_{\mathrm{k},\mathrm{n}}\left(\mathrm{t}\right)=\kappa \sqrt{{\mathrm{p}}_{\mathrm{k},\mathrm{n}}}{\beta }_{\mathrm{k},\mathrm{n}}{\mathrm{e}}^{\mathrm{j}2\pi {\mu }_{\mathrm{k},\mathrm{n}}\mathrm{t}}\mathbf{v}\left({\theta }_{\mathrm{k},\mathrm{n}}\right){\mathbf{u}}^{\mathrm{H}}\left({\theta }_{\mathrm{k},\mathrm{n}}\right){\mathbf{f}}_{\mathrm{k},\mathrm{n}}{\mathrm{s}}_{\mathrm{k},\mathrm{n}}\left(\mathrm{t}-{\tau }_{\mathrm{k},\mathrm{n}}\right)+{\mathbf{w}}_{\mathrm{k},\mathrm{n}}\left(\mathrm{t}\right)$$

(9)

2.3. Radar Measurement Model

The radar echo measurement model is associated with the calibration process in the EKF. The RSU continuously calibrates the EKF predictions of the kinematic parameters by exploiting the radar echoes of each time slot. The radar measurement models for linear and curvilinear mobility scenarios are similar, differing only in speed, as will be noted below. Specifically, the Doppler frequency ${\widehat{\mu }}_{k,n}$ and the time-delay ${\widehat{\tau }}_{k,n}$ of the k-th vehicle can be estimated via the conventional matched-filtering [16], i.e., the transmitted signal with delay and Doppler shift is used to perform correlation calculations with the received radar echo signal. The estimated delay and Doppler frequency are acquired by maximizing correlation expression, which can be expressed as

$$\left\{{\widehat{\tau }}_{\mathrm{k},\mathrm{n}},{\widehat{\mu }}_{\mathrm{k},\mathrm{n}}\right\}=\underset{{}_{\tau ,\mu }}{argmax}{\left|{\int }_0^{\Delta \mathrm{T}}{\mathbf{r}}_{\mathrm{k},\mathrm{n}}\left(\mathrm{t}\right){\mathrm{s}}_{\mathrm{k},\mathrm{n}}^{*}(\mathrm{t}-\tau ){\mathrm{e}}^{-\mathrm{j}2\pi \mu \mathrm{t}}\mathrm{dt}\right|}^2$$

(10)

By exploiting the estimated time-delay ${\widehat{\tau }}_{k,n}$ and Doppler frequency ${\widehat{\mu }}_{k,n}$, the received radar echo at the RSU can be compensated, which is given by

$$\begin{array}{c}{\int }_0^{\Delta \mathrm{T}}{\mathbf{r}}_{\mathrm{k},\mathrm{n}}\left(\mathrm{t}\right){\mathrm{s}}_{\mathrm{k},\mathrm{n}}^{*}\left(\mathrm{t}-{\widehat{\tau }}_{\mathrm{k},\mathrm{n}}\right){\mathrm{e}}^{-\mathrm{j}2\pi {\widehat{\mu }}_{\mathrm{k},\mathrm{n}}\mathrm{t}}\mathrm{dt}\hfill \\ =\kappa \sqrt{{\mathrm{p}}_{\mathrm{k},\mathrm{n}}}{\beta }_{\mathrm{k},\mathrm{n}}\mathbf{v}\left({\theta }_{\mathrm{k},\mathrm{n}}\right){\mathbf{u}}^{\mathrm{H}}\left({\theta }_{\mathrm{k},\mathrm{n}}\right){\mathbf{f}}_{\mathrm{k},\mathrm{n}}\hfill \\ ·{\int }_0^{\Delta \mathrm{T}}{\mathrm{s}}_{\mathrm{k},\mathbf{n}}\left(\mathrm{t}-{\tau }_{\mathrm{k},\mathrm{n}}\right){\mathrm{s}}_{\mathrm{k},\mathrm{n}}^{*}\left(\mathrm{t}-{\widehat{\tau }}_{\mathrm{k},\mathrm{n}}\right){\mathrm{e}}^{\mathrm{j}2\pi \left({\mu }_{\mathrm{k},\mathrm{n}}\mathrm{t}-{\widehat{\mu }}_{\mathrm{k},\mathrm{n}}\mathrm{t}\right)}\mathrm{dt}\hfill \\ +{\int }_0^{\Delta \mathrm{T}}{\mathbf{w}}_{\mathrm{k},\mathrm{n}}\left(\mathrm{t}\right){\mathrm{s}}_{\mathrm{k},\mathrm{n}}^{*}\left(\mathrm{t}-{\widehat{\tau }}_{\mathrm{k},\mathrm{n}}\right){\mathrm{e}}^{\mathrm{j}2\pi {\widehat{\mu }}_{\mathrm{k},\mathrm{n}}\mathrm{t}}\mathrm{dt}\hfill \\ =\kappa \sqrt{{\mathrm{p}}_{\mathrm{k},\mathrm{n}}}\sqrt{G}{\beta }_{\mathrm{k},\mathrm{n}}\mathbf{v}\left({\theta }_{\mathrm{k},\mathrm{n}}\right){\mathbf{u}}^{\mathrm{H}}\left({\theta }_{\mathrm{k},\mathrm{n}}\right){\mathbf{f}}_{\mathrm{k},\mathrm{n}}+{\tilde{\mathbf{w}}}_{\theta }\hfill \end{array}$$

(11)

where G denotes matched-filtering gain. We normalize (11) to acquire the measurement model of the angle ${\theta }_n$ and reflection coefficient ${\beta }_n$ as follows,

$$\begin{array}{c}{\tilde{\mathbf{r}}}_{\mathrm{k},\mathrm{n}}=\kappa {\beta }_{\mathrm{k},\mathrm{n}}\mathbf{v}\left({\theta }_{\mathrm{k},\mathrm{n}}\right){\mathbf{u}}^{\mathrm{H}}\left({\theta }_{\mathrm{k},\mathrm{n}}\right)\mathbf{u}\left({\widehat{\theta }}_{\mathrm{k},\mathrm{n}|\mathrm{n}-1}\right)+{\mathbf{w}}_{\theta }\end{array}$$

(12)

where ${\mathbf{w}}_{\theta }$ denotes the measurement noise normalized by the transmit power $p_{k,n}$ and the matched-filtering gain G, with zero mean and variance of ${\sigma }_1^2$. Accordingly, the distance $d_n$ and velocity $v_n$ also have measurement models, which yields

$$\begin{array}{c}{\tau }_{\mathrm{k},\mathrm{n}}=\frac{2{\mathrm{d}}_{\mathrm{k},\mathrm{n}}}{\mathrm{c}}+{\mathrm{w}}_{\tau }\end{array}$$

(13)

$$\begin{array}{c}{\mu }_{\mathrm{k},\mathrm{n}}=\frac{2{\mathrm{v}}_{\mathrm{k},\mathrm{n}}\cos {\theta }_{\mathrm{k},\mathrm{n}}{\mathrm{f}}_{\mathrm{c}}}{\mathrm{c}}+{\mathrm{w}}_{\mu }\end{array}$$

(14)

It should be noted that for curvilinear mobility scenario, $v_{k,n}=\sqrt{v_{x_n}^2+v_{y_n}^2}$, i.e.,

$${\mu }_{k,n}=\frac{2\sqrt{v_{x_n}^2+v_{y_n}^2}cos\left({\theta }_{k,n}\right)f_c}{c}+{\mathrm{w}}_{\mu }$$

(15)

where c and $f_c$ denote the speed of light and the carrier frequency, respectively. $w_{\tau }$ and $w_{\mu }$ denote the Gaussian noise which has a mean of zero and a variance of ${\sigma }_2^2$ and ${\sigma }_3^2$, respectively. It should be noted that ${\sigma }_1^2$, ${\sigma }_2^2$, and ${\sigma }_3^2$ are related to the signal-to-noise ratio (SNR) at the RSU [21], which is derived as [16]

$$\begin{array}{c}{\sigma }_1^2=\frac{{\sigma }^2}{\mathrm{G}{\mathrm{p}}_{\mathrm{k},\mathrm{n}}},{\sigma }_{\mathrm{i}}^2=\frac{{\mathrm{a}}_{\mathrm{i}}^2{\sigma }^2}{\mathrm{G}{\kappa }^2{\left|{\beta }_{\mathrm{k},\mathrm{n}}\right|}^2{\left|{\mathbf{u}}^{\mathrm{H}}\left({\theta }_{\mathrm{k},\mathrm{n}}\right)\mathbf{v}\left({\widehat{\theta }}_{\mathrm{k},\mathrm{n}|\mathrm{n}-1}\right)\right|}^2{\mathrm{p}}_{\mathrm{k},\mathrm{n}}},i=2,3\end{array}$$

(16)

where $a_1$, $a_2$, $a_3$ are constants related to system configuration.

2.4. Communication Signal Model

At time slot n, the k-th vehicle formulates receiving beam to receive the downlink signal from the RSU, which can be expressed as

$$\begin{array}{c}{\mathrm{c}}_{\mathrm{k},\mathrm{n}}\left(\mathrm{t}\right)=\sqrt{{\mathrm{p}}_{\mathrm{k},\mathrm{n}}}{\alpha }_{\mathrm{k},\mathrm{n}}{\mathbf{w}}_{\mathrm{k},\mathrm{n}}^{\mathrm{H}}{\mathbf{H}}_{\mathrm{k},\mathrm{n}}{\mathbf{f}}_{\mathrm{k},\mathrm{n}}{\mathrm{s}}_{\mathrm{k},\mathrm{n}}\left(\mathrm{t}\right)+{\mathrm{n}}_{\mathrm{c}}\left(\mathrm{t}\right)\end{array}$$

(17)

where ${\alpha }_{k,n}=\tilde{\alpha }{e}^{j\frac{2\pi }{\lambda }{d}_{k,n}}$ denotes the channel coefficient during communication, in which $\tilde{\alpha }$ is path loss, $\frac{2\pi }{\lambda }{d}_{k,n}$ is the phase of the channel. ${\mathbf{H}}_{k,n}$ denotes the mmWave channel between RSU and vehicle. ${\mathbf{w}}_{k,n}^H$ and ${\mathbf{f}}_{k,n}$ denote beam combining and precoding vector, and are similarly defined as in (8) with $N_t$ and M antennas, respectively. $n_c\left(t\right)$ denotes the Gaussian noise which has a mean of zero and a variance of ${\sigma }_c^2$.

Specifically, path loss $\tilde{\alpha }$ refers to 3GPP TR 38.901 [22], which presents a path loss model of rural macrocellular (RMa) scenario including LoS and Non-Line of Sight (NLoS) path. The path loss is calculated as follows

$$\begin{array}{c}PL=P{r}_{LOS}P{L}_{RMa-LOS}+\left(1-P{r}_{LOS}\right)P{L}_{RMa-\mathrm{N}LOS}\end{array}$$

(18)

where ${Pr}_{LOS}$ denotes the probability of LoS path, ${PL}_{RMa-LOS}$ and ${PL}_{RMa-LOS}$ denote the loss of the LoS path and NLoS path, respectively. The specific calculations are detailed in Appendix A.1.

It is worth noting that ${\mathbf{w}}_{k,n}^H$ is formulated by AOA at the vehicle side which AOA is the same as AOD in linear mobility scenario, while AOA is different from AOD in curvilinear mobility scenario. Thus, we employ the classical MUltiple SIgnal Classification (MUSIC) algorithm which has superior performance in angular resolution to estimate AOA in curvilinear mobility scenarios at the vehicle side [23].

${\mathbf{H}}_{k,n}$ is a wideband eSV channel containing an LoS path, where Rice factor is introduced to control the power ratio between the LoS and NLoS paths. Moreover, the time-delay of each path is different, thus ensuring that the channel is wideband. The wideband eSV channel used in the article is given by

$$\begin{array}{c}{\mathbf{H}}_{\mathbf{k},\mathbf{n}}={\displaystyle \sum _{j=1}^{N_{cl}}}{\displaystyle \sum _{i=1}^{N_{ray}}}{a}_{k,n,j,i}{e}^{j2\pi e_{k,n,j,i}t}\sqrt{\frac{N_{t}M}{N_{cl}{N}_{ray}+1}}\mathbf{b}\left({\theta }_{k,n,j,i}^{rx}\right){\mathbf{a}}^H\left({\theta }_{k,n,j,i}^{tx}\right)\sqrt{\frac{1}{K_{rice}+1}}\hfill \\ +a_{k,n}^{LoS}{e}^{j2\pi e_{k,n}^{LoS}t}\sqrt{\frac{N_{t}M}{N_{cl}{N}_{ray}+1}}\mathbf{b}\left({\theta }_{k,n}^{LoS,rx}\right){\mathbf{a}}^H\left({\theta }_{k,n}^{LoS,tx}\right)\sqrt{\frac{K_{rice}}{K_{rice}+1}}\hfill \end{array}$$

(19)

where $N_{cl}$ is the number of clusters, $N_{ray}$ is the number of paths in each cluster, and therefore the total number of paths of NLoS is $N_{cl}{N}_{ray}$. $a_{k,n,j,i}$ and ${\varrho }_{k,n,j,i}=\frac{v_{k,n}cos{\theta }_{k,n,j,i}{f}_c}{c}$ denote complex gain and Doppler frequency shift of the k-th vehicle for the i-th path in the j-th cluster at the n-th slot, respectively. $N_t$ is the number of transmitting antennas at the RSU, while M is the number of receiving antennas at the vehicle. $\mathbf{a}\left(\theta \right)$ and $\mathbf{b}\left(\theta \right)$ denote array antenna response at the RSU and the vehicle, and are similarly defined as in (6) and (7) with $N_t$ and M antennas, respectively. ${\theta }_{k,n,j,i}^{rx}$ and ${\theta }_{k,n,j,i}^{tx}$ denote the AOD and AOA of the k-th vehicle for the i-th path in the j-th cluster at the n-th slot. $K_{rice}$ is the Rice factor. Correspondingly, $a_{k,n}^{LoS}$, ${\varrho }_{k,n}^{LoS}$, ${\theta }_{k,n}^{LoS,rx}$, ${\theta }_{k,n}^{LoS,tx}$ are complex gain, Doppler frequency shift, AOA, AOD of the LoS path.

Assume that $s_{k,n}\left(t\right)$ has unit power, then the received SNR for the k-th vehicle at the n-th slot is expressed as

$$\begin{array}{c}\mathrm{SN}{\mathrm{R}}_{\mathrm{k},\mathrm{n}}=\frac{{\mathrm{p}}_{\mathrm{k},\mathrm{n}}{\alpha }_{\mathrm{k},\mathrm{n}}{\left|{\mathbf{w}}_{\mathrm{k},\mathrm{n}}^{\mathrm{H}}{\mathbf{H}}_{k,n}{\mathbf{f}}_{\mathrm{k},\mathrm{n}}\right|}^2}{{\sigma }_{\mathrm{C}}^2}\end{array}$$

(20)

Correspondingly, the sum rate of K vehicles is given by

$$\begin{array}{c}{\mathrm{R}}_{\mathrm{n}}={\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{K}}}\mathrm{lo}{\mathrm{g}}_2\left(1+\mathrm{SN}{\mathrm{R}}_{\mathrm{k},\mathrm{n}}\right)\end{array}$$

(21)

3. Radar-Assisted Multi-Rsu Cooperative Beam Tracking

In this paper, we aim to improve beam tracking accuracy by multi-RSU collaboration in a radar-assisted V2I downlink scenario. In this section, we first give the design of the new frame structure and the procedure of beam tracking (Section 3.1). Then we give the specific two steps of beam tracking, which are the derivation of the extended Kalman filter (Section 3.2) and the multi-RSU cooperation procedure (Section 3.3).

3.1. Frame Structure Design and Beam Tracking Procedure

As shown in Figure 3, the DFRC data stream is divided into multiple time slots which correspond to different optimal transmit-receive beam pairs. In each time slot, only when the beam of the RSU and the vehicle are aligned to each other, can the downlink quality-of-service (QoS) be guaranteed. In each time slot, the DFRC signal is partially reflected by the vehicle and partially received at the vehicle, therefore sensing and communication tasks are executed in parallel. In time slot 1, some initial access processes need to be performed first, i.e., the RSU completes the channel measurement by transmitting the downlink pilot and receiving the uplink feedback and thus obtains the initial AOD and AOA. Then the RSU transmits the DFRC signal and the vehicle receives the signal by using the optimal transmit-receive beam pair, followed by the vehicle for uplink feedback transmission. In parallel, the RSU receives radar echoes with a time delay from the vehicle and uploads them to the ES, where AOD prediction based on multi-RSU cooperation is performed. The prediction will then assist in the calculation of beamforming vectors for future moments.

More specifically, each RSU uses the radar echoes to calibrate the predicted kinematic parameters according to the EKF algorithm, then all the RSUs will report their calibration results to ES, where the more accurate vehicle position is obtained based on the reported kinematic parameters and the weights of each RSU. Starting from the second time slot, the transmit beamforming design no longer requires downlink pilot and uplink feedback and is based on cooperative AOD prediction. However, for the receive beam design in a linear mobility scenario, a two-steps prediction will be performed in time slot 1, i.e., the AOD of time slot 2 and time slot 3 will be predicted, where the AOD prediction of time slot 2 is used for the transmit beam design of time slot 2 and the AOD prediction of time slot 3 will be included in the DFRC signal sent in time slot 2 for the formulation of the vehicle’s receive beam in time slot 3. For curvilinear mobility scenarios, AOD and AOA are different, and thus MUSIC algorithm is adopted to estimate AOA and formulate the receive beam. The process of the subsequent time slot is the same as that of the previous time slot.

As shown in Figure 4, beam tracking procedure is divided into two steps. In step I, we employ the EKF algorithm to track vehicle kinematic parameters and report them to the ES, while we have designed two scenarios which are vehicle linear and curvilinear mobility, respectively. In step II, ES calculates the optimal predicted position of the vehicle through the results reported by each RSU and their respective weights and sends the prediction results to each RSU for the input of EKF at the next slot. In the following, we will give a more detailed description of these two steps.

3.2. Extend Kalman Filtering for Linear and Curvilinear Mobility Scenarios

In this subsection, we propose an EKF scheme for beam prediction and tracking. Firstly, kinematic state variables at the $(n-1)$-th slot are used to predict those at the n-th slot based on the kinematic evolution model. Then, the predicted AOD is exploited to formulate a transmitted beam pointing towards the vehicle and thereby transmitting the DFRC signal. The partial signal is reflected by the vehicle and arrives at the RSU side, which is used to calibrate the kinematic state variables based on the radar measurement model. Normally, calibrated state variables are used as inputs directly for the EKF prediction at the next time slot, however, we reduce the gap between calibration results and true kinematic parameters of the vehicle by adding multi-RSU cooperation. Through a process of iterative prediction of kinematic parameters and beam tracking, the RSU can perform simultaneous sensing and communication with the vehicle.

The kinematic evolution model and radar measurement model are needed to enable EKF prediction and calibration, which are given by

$$\left\{\begin{array}{c}KinematicEvolutionModel:{\mathbf{x}}_{\mathrm{n}}=\mathbf{p}\left({\mathrm{x}}_{\mathrm{n}-1}\right)+{\mathbf{q}}_{\mathrm{n}}\hfill \\ RadarMeasurementModel:{\mathbf{y}}_{\mathrm{n}}=\mathbf{f}\left({\mathrm{x}}_{\mathrm{n}}\right)+{\mathbf{w}}_{\mathrm{n}}\hfill \end{array}\right.$$

(22)

where $x_{n-1}$ is a kinematic state variable at time slot $n-1$, $\mathbf{p}\left(\right)$ and $\mathbf{f}\left(\right)$ denote compact forms of predictive and measurement models, respectively. ${\mathbf{q}}_n$ and ${\mathbf{w}}_n$ represent predicting and measuring Gaussian noise with zero mean and corresponding variance.

For linear mobility scenario, according to [16], $x={\left[\theta ,d,v,\beta \right]}^T$, $y={\left[{\tilde{r}}^T,\tau ,\mu \right]}^T$, $\mathbf{p}\left(\right)$ is defined in (1), with $\mathbf{q}={\left[q_{\theta },q_d,q_v,q_{\beta }\right]}^T$, $\mathbf{f}\left(\right)$ is defined as (12)–(14), with $\mathbf{w}={\left[w_{\theta }^T,w_{\tau },w_{\mu }\right]}^T$. Thus, the variance matrix in the EKF is expressed by

$${\mathbf{Q}}_s=diag\left({\sigma }_{\theta }^2,{\sigma }_d^2,{\sigma }_v^2,{\sigma }_{\beta }^2\right)$$

(23)

$${\mathbf{Q}}_{\mathrm{m}}=diag\left({\sigma }_1^2{1}_{{\mathrm{N}}_{\mathrm{r}}}^{\mathrm{T}},{\sigma }_2^2,{\sigma }_3^2\right)$$

(24)

where ${\mathbf{1}}_{N_r}$ denotes a column vector consisting of all ones with a length of $N_r$.

For curvilinear mobility scenario, on the top of [16], $x={\left[\theta ,d,v_y,v_x,\beta \right]}^T$, $y={\left[{\tilde{r}}^T,\tau ,\mu \right]}^T$, $\mathbf{p}\left(\right)$ is defined in (3), with $\mathbf{q}={\left[q_{\theta },q_d,q_{v_y},q_{v_x},q_{\beta }\right]}^T$, $\mathbf{f}\left(\right)$ is defined as (12)–(15), with $\mathbf{w}={\left[w_{\theta }^T,w_{\tau },w_{\mu }\right]}^T$. Thus, the variance matrix in the EKF is expressed by

$${\mathbf{Q}}_s=diag\left({\sigma }_{\theta }^2,{\sigma }_d^2,{\sigma }_{v_y}^2,{\sigma }_{v_x}^2,{\sigma }_{\beta }^2\right)$$

(25)

$${\mathbf{Q}}_{\mathrm{m}}=diag\left({\sigma }_1^2{1}_{{\mathrm{N}}_{\mathrm{r}}}^{\mathrm{T}},{\sigma }_2^2,{\sigma }_3^2\right)$$

(26)

We have acquired the kinematic evolution model and radar measurement model which can enable the EKF technique. In accordance with the typical process of Kalman filtering [21], the kinematic parameters prediction and calibration are given by

(1)

Predict Kinematic State Variable

$$\begin{array}{c}{\widehat{\mathbf{x}}}_{n|n-1}=\mathbf{p}\left({\widehat{\mathbf{x}}}_{n-1}\right),{\widehat{\mathbf{x}}}_{n+1|n-1}=\mathbf{p}\left({\widehat{\mathbf{x}}}_{n|n-1}\right)\end{array}$$

(27)

(2)

Linearize Kinematic Evolution Model and Radar Measurement Model

$$\begin{array}{c}{\mathbf{P}}_{n-1}=\frac{\partial \mathbf{p}}{\partial \mathbf{x}}{|}_{\mathbf{x}={\widehat{\mathbf{x}}}_{n-1}},{\mathbf{F}}_n=\frac{\partial \mathbf{f}}{\partial \mathbf{x}}{|}_{\mathbf{x}={\widehat{\mathbf{x}}}_{n|n-1}}\end{array}$$

(28)

(3)

Predict MSE Matrix

$$\begin{array}{c}{\mathbf{M}}_{n|n-1}={\mathbf{P}}_{n-1}{\mathbf{M}}_{n-1}{\mathbf{P}}_{n-1}^H+{\mathbf{Q}}_s\end{array}$$

(29)

(4)

Calculate Kalman Gain

$$\begin{array}{c}{\mathbf{K}}_n={\mathbf{M}}_{n|n-1}{\mathbf{F}}_n^H{\left({\mathbf{Q}}_m+{\mathbf{F}}_n{\mathbf{M}}_{n|n-1}{\mathbf{F}}_n^H\right)}^{-1}\end{array}$$

(30)

(5)

Calibrate Kinematic State Variable

$$\begin{array}{c}{\widehat{\mathbf{x}}}_n={\widehat{\mathbf{x}}}_{n|n-1}+{\mathbf{K}}_n\left({\mathbf{y}}_n-\mathbf{f}\left({\widehat{\mathbf{x}}}_{n|n-1}\right)\right)\end{array}$$

(31)

(6)

Update MSE Matrix

$$\begin{array}{c}{\mathbf{M}}_n=\left(\mathbf{I}-{\mathbf{K}}_n{\mathbf{F}}_n\right){\mathbf{M}}_{n|n-1}\end{array}$$

(32)

Where step (2) is related to the Jacobi matrix, which is derived in detail in Appendix A.2.

3.3. Multi-RSU Cooperative Beam Tracking

Multi-RSU cooperation can address the problems of low localization accuracy and poor coverage in the single-RSU network. Since the sensing performance of each RSU is related to the relative position of the vehicle to it, various RSUs can provide different position predictions of vehicles which could be jointly exploited to improve the accuracy of kinematic parameter prediction.

At the n-th time slot, the L RSUs receive K echo signals reflected from K vehicles, which are exploited to formulate measurements ${\mathbf{y}}_{k,l,n}$, $k=1,2,...,K,l=1,2,\dots ,L$, where K and L denote the number of vehicles and RSUs, respectively. Since kinematic state prediction ${\widehat{\mathbf{x}}}_{k,l,n|n-1}$ and its corresponding measurement ${\mathbf{y}}_{k,l,n}$ have a high correlation, we calculate the measurement predictions of L RSUs given K kinematic state predictions ${\widehat{\mathbf{x}}}_{p,l,n|n-1},p=1,2,\dots ,K$ at n-th time slot, which can be expressed as

$${\widehat{\mathbf{y}}}_{p,l,n}=\mathbf{f}\left({\widehat{\mathbf{x}}}_{p,l,n|n-1}\right),p=1,2,\dots ,K$$

(33)

Before each RSU reports prediction to ES, echoes from different vehicles need to be differentiated. Each RSU calculates the Euclidean distance between ${\mathbf{y}}_{k,l,n}$ and each of the K measurement predictions ${\widehat{\mathbf{y}}}_{p,l,n}$, thereby associating radar echoes measurements ${\mathbf{y}}_{k,l,n}$ with the corresponding vehicles based on the smallest Euclidean distance, which can be given by

$$p=arg\underset{p}{min}{\mathbf{y}}_{k,l,n}-{\widehat{\mathbf{y}}}_{p,l,n},\mathrm{p}=1,2,\dots ,K$$

(34)

As for multi-RSU cooperation, the ES first obtains different position prediction coordinates of the vehicle based on the kinematic state variables calibrated by EKF reported by each RSU, which can be expressed as

$${{\widehat{x}}^{pre}}_{k,l,n}={\widehat{d}}_{k,l,n}·cos{\widehat{\theta }}_{k,l,n}$$

(35)

$${{\widehat{y}}^{pre}}_{k,l,n}={\widehat{d}}_{k,l,n}·sin{\widehat{\theta }}_{k,l,n}$$

(36)

where ${\widehat{d}}_{k,l,n}$ and ${\widehat{\theta }}_{k,l,n}$ denote the distance and angle of k-th vehicle reported by the l-th RSU at the n-th time slot, respectively. ${\widehat{x}}_{k,l,n}^{pre}$ denotes the calculated x-coordinate of the vehicle based on ${\widehat{d}}_{k,l,n}$ and ${\widehat{\theta }}_{k,l,n}$. Similarly for the y-coordinate of the vehicle ${\widehat{y}}_{k,l,n}^{pre}$.

Since the distance between the RSU and the vehicle will affect the tracking performance of the RSU, the prediction accuracy varies significantly for RSUs distributed in different geographical locations. Therefore, it is necessary to design the weights corresponding to different RSUs during cooperation. In our paper, the weight of each RSU is related to the Euclidean distance between the real and predicted echoes. The weights are calculated as follows

$$E_{k,l,n}=\underset{p}{min}{y}_{k,l,n}-{\widehat{y}}_{p,l,n},\mathrm{p}=1,2,\dots ,K$$

(37)

$$w_{k,l,n}=\frac{1}{L-1}\left(1-\frac{E_{k,l,n}}{{\sum }_{i=1}^L{E}_{k,i,n}}\right)$$

(38)

where $E_{k,l,n}$ denotes the smallest Euclidean distance which is associated with the accuracy of RSU beam tracking. Smaller Euclidean distances represent more accurate position prediction which leads to effective beam tracking. $w_{k,l,n}$ denotes the weight of each RSU involved in the cooperation, while smaller Euclidean distances or more accurate predictions of kinematic parameters are associated with larger weights. $1-\frac{E_{k,l,n}}{{\sum }_{i=1}^L{E}_{k,i,n}}$ denotes the prediction confidence of the l-th RSU, which is positively correlated with the weight of cooperation. L denotes the total number of RSUs involved in the cooperation. As for the limitation of Equation (38), the cooperation weight $w_{k,l,n}$ is limited between zero and one, and the sum of the weights of all RSUs is limited to one by using the term $\frac{1}{L-1}$, which thus guarantees the effectiveness of the cooperation results. In addition, there are no other limitations on the use of Equation (38). As a result, the coordinates of the optimal predicted vehicle position based on multi-RSU cooperation are expressed as

$${{\widehat{x}}^{co}}_{k,n}=\sum _{l=1}^L{w}_{k,l,n}{{\widehat{x}}^{pre}}_{k,l,n}$$

(39)

$${{\widehat{y}}^{co}}_{k,n}=\sum _{l=1}^L{w}_{k,l,n}{{\widehat{y}}^{pre}}_{k,l,n}$$

(40)

where ${\widehat{x}}_{k,n}^{co}$ denotes the x-coordinate predicted by ES for vehicle k at time slot n, similarly for the y-coordinate. Cooperation takes advantage of distributed sensing to bring the predicted location closer to the real vehicle location. The optimal distance and AOD between the l-th RSU and k-th vehicle predicted by ES at time slot n are given by

$${\widehat{d}}_{k,l,n}^{co}=\sqrt{{\left({\widehat{x}}_{k,n}^{co}-x_{R_l}\right)}^2+{\left({\widehat{y}}_{k,n}^{co}-y_{R_l}\right)}^2}$$

(41)

$${\widehat{\theta }}_{k,l,n}^{co}={tan}^{-1}\left(\frac{{\widehat{y}}_{k,n}^{co}-y_{R_l}}{{\widehat{x}}_{k,n}^{co}-x_{R_l}}\right)$$

(42)

where ${x_R}_l$ and ${y_R}_l$ denote x-coordinate and y-coordinate of the l-th RSU. ${\widehat{d}}_{k,l,n}^{co}$ and ${\widehat{\theta }}_{k,l,n}^{co}$ are more accurate kinematic state variables that are used as input for the EKF prediction at the next time slot.

Since multiple RSUs are considered in this paper, switching between RSUs also needs to be addressed. We propose three switching strategies as follows

• Distance Priority: The RSU with the closest distance to the vehicle is selected as the serving RSU. Although this strategy has a low overhead, its spectral efficiency is also very low.
• Reference Signal Receiving Power (RSRP) Priority: Multiple RSUs perform small-range beam scanning separately, and the RSU corresponding to the maximum received RSRP at the vehicle is used as the serving RSU. However, this strategy provides high spectrum efficiency but also a high overhead.
• Radar Echo Strength Priority: The RSU with the highest received radar echo strength is selected as the serving RSU. This strategy avoids beam scanning and thus reduces overhead, and also has high spectral efficiency.

4. Numerical Results

In this section, we conduct simulations in a radar-assisted multi-RSU cooperation V2I network under the mmWave channel and provide extensive numerical results to verify the effectiveness of the proposed method. In the simulated V2I network, each of the L RSUs equipped with $N_t$ transmit antennas and $N_r$ receives antennas to serve K M-antenna vehicles, while all RSUs are linked to ES as defined in the section II system model. Figure 5 illustrates the relative positions of the L RSUs and the K vehicles by the two-dimensional (2D) coordinate system for linear and curvilinear mobility scenarios, respectively. In a linear mobility scenario, 3 RSUs and 2 vehicles are considered, while the coordinates of the 3 RSUs are [300 m, 70 m], [260 m, 70 m], [280 m, 130 m] and the initial coordinates of the 2 vehicles are [340 m, 100 m], [330 m, 100 m], respectively. Similarly, in a curvilinear mobility scenario, 2 RSUs with the position coordinates [290 m, 230 m], [200 m, 270 m] and 2 vehicles with the initial position coordinate [300 m, 300 m], [290 m, 290 m] are considered, as shown in Figure 5. The other default parameters are the same for both scenarios, which are given by

Table 1. Default settings in simulations.

Parameter	Value
Carrier frequency	60 GHz
Channel model	eSV (9 paths)
Rice factor	2
Complex RCS	25 + 25i
Bandwidth	200 MHz
Downlink transmit power	20 dBm
Noise power spectral density	−174 dBm/Hz
$\Delta T$	0.018 s
$a_1$	1
$a_2$	$6.7\times {10}^{-7}$
$a_3$	$2\times {10}^4$
G	10

.

4.1. Performance for Linear Mobility Scenario

In linear mobility scenario, the vehicle moves in a straight line at a nearly constant speed. We first evaluate the sensing and communication performances of the proposed multi-RSU cooperation technique for a single vehicle, i.e., 3 RSUs cooperate with each other to track a vehicle. For the kinematic state evolution noises, we set ${\sigma }_{\theta }=0.4°$, ${\sigma }_d=1$ m, ${\sigma }_v=1$ m/s and ${\sigma }_{\beta }=0.4$.

To evaluate the beamforming performance, we investigate the tracking of AOD during vehicle motion of the V2I network by EKF. Figure 6 presents the tracking performance of AOD per RSU versus the sampling point, both cooperative and non-cooperative. It is shown that the prediction accuracy of AOD can be improved by multi-RSU cooperation, which is closer to the real physical angle. Multi-RSU cooperation integrates the prediction of RSUs with different sensing performances and improves the vehicle location prediction accuracy through distributed sensing, which in turn can improve the AOD tracking effect. As a comparison, the single RSU is easily affected by channel variations and noise. The predicted AOD is likely to deviate from the real AOD when the vehicle is close to the RSU, i.e., in the interval where the AOD changes rapidly. Correspondingly, in Figure 7, we demonstrate the results for radar sense in terms of root mean squared error (RMSE) for both cooperative and non-cooperative AOD tracking. Simulation results show that the angular error predicted by the EKF without multi-RSU cooperation is particularly large and fails to correctly sense the real-time vehicle position.

To evaluate the communication performance, Figure 8 presents the received SNR at the vehicle side versus the sampling point. Simulation results show that the variation of the received SNR of the vehicle with multi-RSU cooperation is stable, while that of the vehicle without cooperation is fluctuating drastically. The SNR is very related to the direction of the transmitted and received beams, and a higher SNR can only be achieved when both the RSU and the vehicle accurately direct their narrow beams toward each other’s direction. Therefore, beamforming based on accurate AOD tracking is a guarantee for higher received SNR. Moreover, in the linear mobility scenario, we assume that the AOD and AOA are equal, so the precision of AOD tracking affects both the transmit and receive beams, which in turn influences the received SNR even more.

In Figure 9, the spectral efficiency of different RSU switching methods versus the downlink transmit power is shown. Simulation results show that spectral efficiency is significantly improved when multi-RSU cooperation is employed. Specifically, the proposed multiple base station cooperation improves the spectral efficiency by at least 34% over the non-cooperative scheme when the vehicle is in linear mobility, at a transmit power of 20 dBm, regardless of the RSU switching method. Moreover, with multi-RSU cooperation, both the performance of Distance Priority and Radar Echo Strength Priority schemes are very close to that of the RSRP Priority scheme, i.e., the upper bound of spectral efficiency. On the other hand, all three switching methods exhibit low spectral efficiency when there is no cooperation, among which the Distance Priority scheme performs worst due to the extremely inaccurate vehicle position estimation, while the Radar Echo Strength Priority scheme has a relatively large spectral efficiency owing to the consideration of received power.

In addition, a deep neural network(DNN)-based beam prediction algorithm is evaluated and used as a comparison scheme. Specifically, the DNN used for training consists of one input layer, two hidden layers and one output layer, where the number of neurons in the hidden layer is set to 128 and an activation function is added between every two layers to enhance the model’s nonlinear modeling capability. The input of the DNN is the signal power received by the vehicle from the different RSUs at the previous moment. The output is the index of the optimal served RSU and the optimal beamforming angle (AOD) for the next moment. The RSU and vehicle can use the predicted RSU index and AOD from the DNN model to perform beam tracking in highly dynamic scenarios. Since the predicted metrics have both continuous values such as angle and discrete values such as the RSU index, the preprocessing of the data set is divided into two categories, with mean-variance normalization for continuous values and one-hot coding for discrete values. Moreover, both MSE and cross-entropy are used as loss functions and the Adam algorithm is applied to optimize the neural network. As shown in Figure 9, the spectral efficiency of the DNN prediction-based scheme is higher than that of the no-cooperation scheme, which indicates that the DNN model is able to learn the nonlinear correlation between multiple received signal powers and optimal beam directions. However, the accuracy of DNN prediction of AOD is limited by the types of features used for training and the learning capability of the network, while the massive MIMO-based beamforming system is particularly sensitive to beam angle errors, thus the spectral efficiency of the DNN prediction scheme is lower than that of the multi-RSU cooperation scheme proposed in this paper.

In Figure 10, the performance of the proposed scheme with LOS paths blocked is shown, which gives the variation of its spectral efficiency for different numbers of blocked LOS paths. Specifically, three RSUs are deployed for cooperation, and LOS paths exist between each RSU and the vehicle when there is no blocking. Since the RSU cannot accurately track the vehicle when the LOS path is blocked, which will provide a large error during the multi-RSU cooperation, therefore, it is necessary to recognize which RSU’s LOS path is blocked in the existence of blocking and exclude it from the cooperation. Considering that the reflection coefficients of obstacles and vehicles are different, the radar echo strength reflected by obstacles is lower, thus the radar echo strength is used to distinguish the blocked RSUs. The evaluation results are shown in Figure 10. When the LOS paths of all 3 RSUs are not blocked, the multi-RSU cooperation exploits the sensing information provided by all RSUs and thus has the optimal performance. When 1 LOS path is blocked, there are only 2 RSUs involved in cooperation, however, the spectral efficiency only slightly decreases, which indicates that 2 RSUs can still provide a large cooperation gain. When 2 LOS paths are blocked, multi-RSU cooperation cannot be performed, so the spectral efficiency is lowest.

4.2. Performance for Curvilinear Mobility Scenario

In a curvilinear mobility scenario, the vehicle moves in a curve that is combined by uniformly accelerated mobility in two directions. We also first study the case of a single vehicle, i.e., 2 RSUs cooperate with each other to track a vehicle. For the kinematic state evolution noises, we set ${\sigma }_{\theta }=0.4°$, ${\sigma }_d=2$ m, ${\sigma }_{v_y}=1$ m/s, ${\sigma }_{v_x}=1$ m/s and ${\sigma }_{\beta }=0.5$. The AOD tracking performance and the spectral efficiency of different RSU switching methods are given in Figure 11, Figure 12 and Figure 13, respectively. The results are similar to the case of the linear mobility scenario. It is worth noting that due to the higher nonlinearity of the vehicle curvilinear kinematic model, the AOD tracking effect will be slightly decreased compared with the linear mobility scenario, but the spectral efficiency is still improved greatly owing to multi-RSU cooperation. Specifically, when the transmit power is 20 dBm, the spectral efficiency is improved by at least 20% over the non-cooperative scheme due to multiple base station cooperation, regardless of the RSU switching method.

Moreover, we compared the computational complexity of four schemes. For the beam traversal search scheme, i.e., the performance upper bound, the computational complexity is O$\left(N_{t}MLKN\right)$, where $N_t$ denotes the number of transmitting antennas at the RSU, M denotes the number of antennas at the vehicle, L and K denote the number of RSUs and vehicles, respectively. N denotes the number of time slots. Similarly, the computational complexity of the proposed EKF-based and multi-RSU cooperation scheme is O$\left(N(LK{n}^2+3LK+K)\right)$, where n denotes the dimensionality of state variables. As a comparison, the computational complexity of the non-cooperation scheme and the DNN prediction-based scheme is O$\left(NLK{n}^2\right)$ and O$\left((N_{train}+N)(n_1{n}_2+n_2{n}_3+n_3{n}_4+n_4)\right)$, where $N_{train}$ denotes the size of the training set, $n_1$ and $n_4$ denote the dimension of input features and output prediction. $n_2$ and $n_3$ denote the number of neurons in the two hidden layers. To make it more intuitive, the computational complexity is summarized in

Table 2. Computational Complexity.

Scheme	Computational Complexity
beam traversal search	O$\left(N_{t}MLKN\right)$
EKF-based and multi-RSU cooperation	O$\left(N(LK{n}^2+3LK+K)\right)$
EKF-based and non-cooperation	O$\left(NLK{n}^2\right)$
DNN prediction	O$\left((N_{train}+N)(n_1{n}_2+n_2{n}_3+n_3{n}_4+n_4)\right)$

.

5. Conclusions

In this paper, we have proposed a radar-assisted predictive beam tracking design for the V2I network by exploiting ISAC and multi-RSU cooperation. Firstly, on top of the linear mobility scenario, we derive the kinematic parameters evolution model and radar measurement model of the curvilinear mobility scenario which can be used in the EKF for angle prediction and thus beam tracking. Then we designed wideband eSV channel modeling in the communication process, which is more compatible with the existing OFDM system, and we use the more general path loss model given by the protocol. Moreover, aiming for improving sensing accuracy and RSU coverage in ISAC systems, we proposed a multi-RSU cooperation technique based on the prediction accuracy of each RSU which takes full advantage of distributed sensing. Since we considered multiple RSUs, we proposed three different RSU switching methods with an analysis of their respective performance. The proposed schemes have been validated with numerical results, which show that whether the vehicle is moving in a straight line or a curve, multi-RSU collaboration can improve the angular tracking accuracy, which in turn improves the beam quality and obtains higher spectral efficiency. In addition, different RSU switching methods have also been proven to be highly effective.