Physical layer secure transmission in the coexistence of V2I and V2V with RIS assistance

Abstract

The security of vehicle communication becomes increasingly important due to the transmission requirement of large-scale security and privacy data in the Internet of Vehicles (IoV). This paper investigates the physical layer secure transmission with reconfigurable intelligent surface (RIS) assistance in the coexistence of vehicle-to-infrastructure (V2I) and vehicle-to-vehicle (V2V). It establishes a communication model for physical layer secure transmission where the eavesdropping user exists. Based on this model, we propose a V2I physical layer secure transmission rate (PLSTR) maximization problem and solve it by employing an alternating optimization algorithm based on generalized Rayleigh entropy and semi-definite relaxation programming so as to obtain optimal V2I base station precoding and RIS reflection coefficient matrix. Simulation results further validate the superiority of the proposed algorithm and analyze the impact of relevant parameters on system performance.

1 Introduction

With the widespread application of 5G communication technology and research on 6G communication technology, future Internet of Vehicles (IoV) will gradually achieve ultra-efficient and ultra-reliable wireless communication. IoV will bear the burden of transmitting large-scale safety-related and privacy-related data, making the security of IoV terminal communication increasingly important. However, there are currently still issues with low security and unreliable continuity in IoV communication under extreme conditions. The current IoV communication mainly adopts two paradigms: vehicle-to-infrastructure (V2I) and vehicle-to-vehicle (V2V). On the one hand, vehicle terminals communicate through base stations using 5G technology; on the other hand, they can also directly communicate with nearby vehicles using device-to-device (D2D) technology. The appropriate paradigm is chosen for communication based on data type and transmission performance requirements. As the scale of IoV expands, eavesdropping terminals may be constantly present, posing challenges to physical layer secure transmission. Therefore, further research is needed to enhance physical layer security spectrum efficiency under the coexistence of V2I and V2V paradigms.

In recent years, the emergence of reconfigurable intelligent surface (RIS) technology has played an important role in promoting mobile communication development, which has brought huge gains in signal enhancement, blind area coverage and sensing. RIS is composed of a large number of electromagnetic units with simple circuits and low power consumption. If regarded as signal reflection function, these units can be flexibly controlled by digital circuits. Through the circuit control of these units, the phase, amplitude or frequency of the incident electromagnetic signal can be adjusted, so that the reflected electromagnetic wave has a directional beamforming. Therefore, RIS is considered as one of the key technologies for future mobile communication due to its advantages of low power consumption, low cost and flexibility. Research on RIS-assisted mobile communication has also been greatly developed. The channel modeling and channel estimation problem assisted by RIS [1, 2], the information and energy synchronous transmission combined with RIS [3, 4], the physical layer security transmission assisted by RIS [5], low-complexity phase shift optimization of imperfect CSI for RIS-assisted massive MIMO [6, 7], and the assisted cognitive radio communication in [8] have been more and more widely and deeply studied.

The ability of RIS to reconfigure the wireless environment in a favorable way to improve the quality of service (QoS) of wireless communication makes RIS a promising candidate for enhancing V2X applications, and the research on RIS-assisted vehicle networking communication has also become one of the hot topics. In IoV, the radio propagation is negatively affected by a large number of objects (such as high-rise buildings) and dynamic vehicles, and there are blind areas where the communication link is blocked within the coverage of the base station. RIS can well assist these blind spots to communicate and enhance the wireless link. The authors of [9, 10] studied the resource allocation problem of RIS-assisted vehicular communication based on slowly varying large-scale fading channel information. By maximizing the total capacity of the V2I link, while ensuring the minimum signal-to-interference plus noise ratio (SINR) of the V2V link, the power allocation, IRS reflection coefficient and spectrum allocation are jointly optimized. As the same object function, the authors of [11] studied a multiuser spectrum sharing problem in RIS-assisted vehicular networks, where multiple V2V links reuse the spectrum already occupied by V2I links. To overcome the difficulty of obtaining instantaneous channel state information (CSI) due to the rapidly varying nature of some V2X channels, this study relies on large-scale (slowly varying) CSI to meet the QoS requirements of V2I and V2V communications. The vehicle transmit power, multiuser detection matrix, V2V link spectral reuse and RIS reflection coefficient are jointly optimized. The authors of [12] studied the use of RIS in millimeter wave IoV. The weighted sum rate maximization problem in the uplink is considered, and in the single-user case, two phase optimization schemes that help reduce the channel estimation overhead are considered. In the multiuser case, the original problem is reformulated into a more convenient form using fractional programming and an alternating optimization algorithm is proposed. [13] further analyzed the possibility of using deep reinforcement learning (DRL) method to solve the above problems and jointly optimized the base station and RIS beamforming to improve the network performance in millimeter wave V2I communication system. Also using the deep reinforcement learning method, [14] formulated the joint optimization problem of roadside units (RSU) resource scheduling and RIS passive beamforming with the objective of maximizing the minimum average transmission rate. A DRL method was used to determine the RSU wireless scheduling, and block coordinate descent (BCD) algorithm was proposed to solve the phase shift matrix. The authors of [15] proposed a three-stage joint resource allocation method for V2I and V2V spectrum sharing. Literature [16] has optimized and solved the spectrum sharing and resource scheduling problem by combining the requirements of low packet delay and high reliable transmission. Literatures [17, 18] further extended the joint beam optimization and resource allocation problem of MISO IoV assisted by RIS to MIMO system. The above studies demonstrate the gain in system throughput brought by RIS-assisted IoV on the basis of reasonable resource allocation and active and passive beams. However, these studies do not consider physical layer security issues. Further research is needed to address RIS-assisted V2X communication in the presence of eavesdropping users.

Other studies extend the performance limits of RIS-assisted IoV communication, including performance analysis, channel estimation and other high-spectrum efficiency access techniques, such as outage probability, average outage time and flat exchange rate of received signal envelope, which has also been studied [19, 20]. In terms of channel estimation, the authors of [21, 22] gave a low overhead channel estimation protocol in RIS-assisted IoV and proposed a compressed sensing channel estimation algorithm based on location information assistance to reduce the additional training overhead of the channel. The performance and joint beam optimization method in the case of imperfect channel estimation were further studied [23]. In order to meet the service requirements of different vehicle communications, the authors of [24] analyzed the sum rate maximization problem under the hybrid OMA/NOMA access scheme and jointly optimized the transmit power allocation, RIS phase shift and vehicle active beamforming under different QoS levels with the constraints of vehicle reliability and rate requirements. In [25, 26], the optimal selection method of multiple RIS was designed based on the maximization of ergodic capacity and the minimization of symbol average error probability. In [27], a conformal RIS deployed on the vehicle body was proposed to mitigate the blocking effect by generating artificial reflections and compensate the non-flat shape of the vehicle body with an appropriate phase pattern. A new conformal RIS design with optimized pre-configuration was proposed to demonstrate its advantages in V2V highway scenarios. However, these have not been analyzed for the physical layer security issues of IoV, and the joint beamforming technology of V2I, V2V and eavesdropping users at the same time with the assistance of RIS has not been studied.

The authors of [28] studied the influence of RIS deployment location and the number of RIS units on the physical layer secure transmission rate (PLSTR) of the IoV system. Literature [26] also studied and compared the PLSTR of RIS with traditional relay in IoV system. In [29], two cases of V2V communication and vehicle-to-RIS communication were considered and closed-form expressions for the outage probability of PLSTR were derived in the presence of one eavesdropper. The potential of using RIS to improve the PLSTR is demonstrated in the above studies. These studies do not comprehensively consider the joint beamforming problem in the presence of V2I and V2V.

Different from the traditional cellular communication, the two communication paradigms of V2I and V2V coexist in IoV communication, and the vehicle terminal movement speed is larger. In this paper, we design a new method to maximize the PLSTR of V2I users in the coexistence of V2I and V2V in the case of existing an eavesdropping vehicle user. Our work is to comprehensively consider the physical layer security problem in the presence of three different types of users, V2I, V2V and eavesdropping vehicles, assisted by RIS. The main contributions in this paper are as follows:

• (1) The physical layer secure communication model of the coexistence of V2I and V2V with the help of RIS is established, and the model of the PLSTR maximization problem of V2I users is given under the constraints of the SINR requirement of V2V users, the RIS phase shift range and the limited transmit power.

• (2) An alternating optimization algorithm based on generalized Rayleigh entropy and semi-definite relaxation program is proposed to jointly optimize the transmission precoding vector at the V2I base station and RIS reflection coefficient matrix to maximize the PLSTR of the V2I user.

• (3) By comparing with the baseline algorithms, the superiority of the proposed algorithm is proved. The simulation results show the gain brought by RIS in the physical layer security transmission of the IoV. The influence of related system parameters on the PLSTR is given.

The rest of this paper is organized as follows. Section 2 describes the methods used in the research process of this paper. Section 3 presents the system and channel model. In Sect. 4, the optimal problem is formulated and the suitable optimization methods are proposed. In Sect. 5, the simulation results are provided to verify the proposed algorithm. Section 6 summarizes the whole paper.

Notations: ${{\varvec{A}}}^{H}$, $\text{Tr}({\varvec{A}})$ and $\text{rank}({\varvec{A}})$ stand for the conjugate transpose, the trace and rank of the matrix ${\varvec{A}}$. $\text{diag}({\varvec{a}})$ stands for taking vector ${\varvec{a}}$ as diagonal elements to form a diagonal matrix. $\widetilde{\text{diag}}({\varvec{A}})$ denotes extracting the diagonal elements of matrix ${\varvec{A}}$ to form a diagonal matrix.

2 Methods/experimental

In this paper investigates, we mainly study the physical layer secure transmission with reconfigurable intelligent surface (RIS) assistance in the coexistence of vehicle-to-infrastructure (V2I) and vehicle-to-vehicle (V2V). We establish a communication model for physical layer secure transmission where the eavesdropping user exists. Based on this model, we formulate a V2I physical layer secure transmission rate (PLSTR) maximization problem which is a coupled non-convex optimization problem. An alternate iterating method is proposed to solve this problem. For the first sub-problem, the generalized Rayleigh entropy is introduced to obtain an analytical solution of the V2I base station precoding. For the second sub-problem, the semi-definite relaxation is used so as to obtain the optimal RIS reflection coefficient matrix. The complexity of the overall algorithm is given. Finally, simulations are carried out by using Matlab, which are further validate the superiority of the proposed algorithm and analyze the impact of relevant parameters on system performance.

3 System model

In this paper, we study the downlink physical layer secure transmission aided by RIS in IoV system where V2I and V2V are coexist, as shown in Fig. 1. There is a V2I vehicle (denoted as C), a base station (denoted as B), a pair of V2V vehicles (denoted as Dt for the message sending vehicle and Dr for the message receiving vehicle), an eavesdropping vehicle (denoted as E), and an RIS reflector plate deployed on the roadside building. The base station is equipped with multiple antennas, which are uniformly and linearly arranged, and the number is denoted as $M$. All other vehicles have single antennas. The number of RIS reflection elements is $N$, and the reflection coefficient of the $n$ th element is ${\phi }_{n}={e}^{j{\theta }_{n}}$, where ${\theta }_{n}\in [0, 2\pi ]$ is the phase shift. In order to reduce the power consumption as much as possible, this paper studies the passive RIS, that is, the amplitude of the RIS reflection coefficient does not exceed 1. This paper assumes the ideal reflection situation, and the amplitude is set to 1, that is, the reflected signal does not attenuate in amplitude, but only changes in phase shift. In fact, in some cases, the system gain may be further improved by using active RIS which can overcome the product-path loss through increasing some power consumption [30]. The vehicle C communicates with the base station B, considering the downlink transmission, and the eavesdropping vehicle E aims to eavesdrop the transmission information from B to C. RIS is deployed on roadside buildings to enhance the incident signal strength and reduce B’s coverage hole. By optimizing the RIS’s phases, the PLSTR of V2I is maximized.

Fig. 1

V2I and V2V downlink transmission of IoV system scenario. A downlink transmission IoV system where V2I and V2V are coexist. A V2I vehicle (denoted as C), a base station (denoted as B), a pair of V2V vehicles (denoted as Dt and Dr), an eavesdropping vehicle (denoted as E), and an RIS reflector plate deployed on the roadside building

It is assumed that the V2V vehicles in the system have completed spectrum resource allocation by the base station and always shares the spectrum with the V2I vehicle for information transmission in a D2D manner. The CSI for this IoV system is assumed to be perfectly known in this paper. The symbol labels of the channels are shown in Table 1, where the symbol ${\mathbb{C}}^{i}$ denotes the set of complex column vectors of size $i$, and the symbol ${\mathbb{C}}^{i\times j}$ denotes the set of complex matrices of size $i\times j$.

Table 1 Channel symbols description

Assume that B sends a symbol $s\in {\mathbb{C}}$ satisfying $\text{E}\left(s{s}^{*}\right)=1$ to C, and define the transmit precoding vector (TPV) ${\varvec{w}}{\in {\mathbb{C}}}^{M\times 1}$; then the signal transmitted from B is ${\varvec{x}}={\varvec{w}}s\in {\mathbb{C}}^{M\times 1}$. Similarly, assume that Dt sends ${s}_{d}\in {\mathbb{C}}$ satisfying $\text{E}\left({s}_{d}{s}_{d}^{*}\right)=1$ to Dr. Since the Dt transmitter is a single transmit antenna, no precoding is required.

Then the received signal of vehicle C is

$${y}_{c}=\left({{\varvec{h}}}_{c}^{\text{H}}+{{\varvec{f}}}_{c}^{H}\boldsymbol{\Phi }{\varvec{H}}\right){\varvec{w}}s+\left({g}_{c}+{{\varvec{f}}}_{c}^{H}\boldsymbol{\Phi }{\varvec{g}}\right){s}_{d}+{n}_{c}$$

(1)

where $\boldsymbol{\Phi }=\text{diag}({\phi }_{1},{\phi }_{2},\cdots ,{\phi }_{N})$ is RIS reflection coefficient matrix (RCM) and ${n}_{c}\sim CN(0,{\sigma }_{c}^{2})$ is Gaussian noise at the receiver of C.

The received signal of the V2V vehicle Dr is

$${y}_{d}=\left({g}_{d}+{{\varvec{f}}}_{d}^{H}\boldsymbol{\Phi }{\varvec{g}}\right){s}_{d}+\left({{\varvec{h}}}_{d}^{H}+{{\varvec{f}}}_{d}^{H}\boldsymbol{\Phi }{\varvec{H}}\right){\varvec{w}}s+{n}_{d}$$

(2)

where ${n}_{d}\sim CN(0,{\sigma }_{d}^{2})$ is Gaussian noise at the receiver of Dr.

The received signal of the eavesdropping vehicle E is

$${y}_{e}=\left({{\varvec{h}}}_{e}^{H}+{{\varvec{f}}}_{e}^{H}\boldsymbol{\Phi }{\varvec{H}}\right){\varvec{w}}s+\left({g}_{e}+{{\varvec{f}}}_{e}^{H}\boldsymbol{\Phi }{\varvec{g}}\right){s}_{d}+{n}_{e}$$

(3)

where ${n}_{e}\sim CN(0,{\sigma }_{e}^{2})$ is Gaussian noise at the receiver of E.

The SINRs of C, Dr and E can be given, respectively, by

$${\text{SINR}}_{c}=\frac{{\left|\left({{\varvec{h}}}_{c}^{H}+{{\varvec{f}}}_{c}^{H}\boldsymbol{\Phi }{\varvec{H}}\right){\varvec{w}}\right|}^{2}}{{\left|{g}_{c}+{{\varvec{f}}}_{c}^{H}\boldsymbol{\Phi }{\varvec{g}}\right|}^{2}+{\sigma }_{c}^{2}}$$

(4)

$${\text{SINR}}_{d}=\frac{{\left|{g}_{d}+{{\varvec{f}}}_{d}^{H}\boldsymbol{\Phi }{\varvec{g}}\right|}^{2}}{{\left|\left({{\varvec{h}}}_{d}^{H}+{{\varvec{f}}}_{d}^{H}\boldsymbol{\Phi }{\varvec{H}}\right){\varvec{w}}\right|}^{2}+{\sigma }_{d}^{2}}$$

(5)

$${\text{SINR}}_{e}=\frac{{\left|\left({{\varvec{h}}}_{e}^{H}+{{\varvec{f}}}_{e}^{H}\boldsymbol{\Phi }{\varvec{H}}\right){\varvec{w}}\right|}^{2}}{{\left|{g}_{e}+{{\varvec{f}}}_{e}^{H}\boldsymbol{\Phi }{\varvec{g}}\right|}^{2}+{\sigma }_{e}^{2}}$$

(6)

Thus, the capacity of vehicle C is ${C}_{c}={\text{log}}_{2}(1+{SINR}_{c})$ bit/s/Hz, and the eavesdropping transmission rate of E is ${C}_{e}={\text{log}}_{2}(1+{SINR}_{e})$ bit/s/Hz. Then the PLSTR of C can be given by

$${C}_{sec}=\text{max}\{0,{C}_{c}-{C}_{e}\}$$

(7)

4 Problem formulation and solving

In this paper, we study the PLSTR maximization problem of vehicle C, which requires that the V2V vehicle communication should meet the minimum SINR requirement, the total transmission power of the multiantenna base station B is limited, and the phase shift range of the RIS unit is limited. In this way, the optimization problem can be formulated as P1.

$$\text{P}1: \underset{{\varvec{w}},\boldsymbol{\Phi }}{\text{max}} {C}_{sec}$$

(8a)

$$\text{s}.\text{t}. {\text{SINR}}_{d}\ge r$$

(8b)

$${\Vert {\varvec{w}}\Vert }^{2}\le {P}_{max}$$

(8c)

$$\left|{\phi }_{n}\right|=1, n=\text{1,2},\cdots ,N$$

(8d)

where the base station TPV ${\varvec{w}}$ and the RIS’s RCM $\boldsymbol{\Phi }$ are two variables to be optimized. Constraint (8b) is the SINR demand of the V2V vehicle, and $r$ is the minimum SINR requirement threshold. The constraint (8c) is the transmit power limitation of base station B, and ${P}_{max}$ is the maximum transmit signal power available to B. The constraint (8d) is the RIS phase shift constraint; since the reflection phase shift range is [0, 2π], the modulus of the reflection coefficient ${\phi }_{n}$ is one.

Problem P1 is a non-convex optimization problem, and the two variables to be optimized are coupled to each other in the objective function (8a) and constraints (8b). This problem is difficult to solve, and it is also difficult to obtain the global optimal solution. In order to decouple, the alternating optimization method can be used to solve the problem iteratively, that is, the RCM $\boldsymbol{\Phi }$ is fixed and the TPV ${\varvec{w}}$ is solved. Then fix TPV and solve RCM. The two steps are iterated alternately until the problem converges and the optimal $\boldsymbol{\Phi }$ and ${\varvec{w}}$ are obtained. Each iteration solves a sub-problem, and each sub-problem may still not be a convex optimization problem. In order to obtain the global optimal solution of the sub-problem, the non-convex problem is usually approximated by a convex optimization problem by using an appropriate mathematical equivalent transformation. This convex approximation method is also used for processing in this paper. The following two subsections 3.1 and 3.2 present alternate solutions for the two variables, respectively.

4.1 Find the optimal TPV

This section finds the solution of TPV ${\varvec{w}}$ under the premise of fixing the RCM $\boldsymbol{\Phi }$. The original problem P1 is then rewritten as problem P2 with only ${\varvec{w}}$ to be optimized.

$$\text{P}2: \underset{\boldsymbol{w }}{\text{max}} {C}_{\text{sec}}$$

$$\text{s}.\text{t}. {\text{SINR}}_{d}\ge r$$

$${\Vert {\varvec{w}}\Vert }^{2}\le {P}_{\text{max}}$$

When the eavesdropping rate is not lower than the transmission rate of the V2I user, that is, when ${C}_{c}-{C}_{e}\le 0$, the V2I user does not transmit information anymore, and ${C}_{\text{sec}}=0$, the base station transmit power is 0 and ${P}_{\text{max}}=0$, then no TPV optimization is performed. Therefore, in what follows, we only consider the case when the eavesdropping rate is lower than the transmission rate of the V2I user, that is, ${C}_{c}-{C}_{e}>0$. When ${C}_{c}-{C}_{e}>0$, ${C}_{\text{sec}}={C}_{c}-{C}_{e}$. In this case, it can be proved that there is a maximum transmission power ${\widetilde{P}}_{\text{max}}$ satisfying the constraints of problem P2, such that ${C}_{\text{sec}}$ is maximum.

In the following, this ${\widetilde{P}}_{\text{max}}$ is found by solving the optimization problem P2-1, where the send precoding to be solved in problem P2-1 is denoted by $\widetilde{{\varvec{w}}}$.

$$\text{P}2-1: \underset{\widetilde{{\varvec{w}}}\boldsymbol{ }}{\text{max}} {\Vert \widetilde{{\varvec{w}}}\Vert }^{2}$$

$$\text{s}.\text{t}. {\left|\left({{\varvec{h}}}_{d}^{H}+{{\varvec{f}}}_{d}^{H}\boldsymbol{\Phi }{\varvec{H}}\right)\widetilde{{\varvec{w}}}\right|}^{2}\le \widetilde{r}$$

$${\Vert \widetilde{{\varvec{w}}}\Vert }^{2}\le {P}_{\text{max}}$$

where $\widetilde{r}={\left|{g}_{d}+{{\varvec{f}}}_{d}^{H}\boldsymbol{\Phi }{\varvec{g}}\right|}^{2}-r{\sigma }_{d}^{2}$. In problem P2-1, the objective function is non-concave. By introducing the slack variable $s$ and letting ${\Vert \widetilde{{\varvec{w}}}\Vert }^{2}\ge s$, the problem P2-1 can be reformulated as P2-2.

$$\text{P}2-2: \underset{\widetilde{{\varvec{w}}} ,\mathit{ s}}{\text{max}} s$$

(9a)

$$\text{s}.\text{t}. {\widetilde{{\varvec{w}}}}^{H}{\left({{\varvec{h}}}_{d}^{H}+{{\varvec{f}}}_{d}^{H}\boldsymbol{\Phi }{\varvec{H}}\right)}^{H}\left({{\varvec{h}}}_{d}^{H}+{{\varvec{f}}}_{d}^{H}\boldsymbol{\Phi }{\varvec{H}}\right)\widetilde{{\varvec{w}}}\le \widetilde{r}$$

(9b)

$${\widetilde{{\varvec{w}}}}^{H}\widetilde{{\varvec{w}}}\le {P}_{\text{max}}$$

(9c)

$${\widetilde{{\varvec{w}}}}^{H}\widetilde{{\varvec{w}}}\ge s$$

(9d)

The constraint (9d) is non-convex, and the first-order Taylor expansion of ${\widetilde{{\varvec{w}}}}^{H}\widetilde{{\varvec{w}}}$ is used to obtain its lower bound $\tilde{\user2{w}}^{H} \tilde{\user2{w}} \ge 2{\text{Re}} \left\{ {\tilde{\user2{w}}^{{(t)^{H} }} \tilde{\user2{w}}} \right\} - \tilde{\user2{w}}^{{(t)^{H} }} \tilde{\user2{w}}^{(t)}$, where ${\widetilde{{\varvec{w}}}}^{\left(t\right)}$ is the result of the iteration of $\widetilde{{\varvec{w}}}$ at the previous time $t$. This lower bound is used to scale the constraint ${\widetilde{{\varvec{w}}}}^{H}\widetilde{{\varvec{w}}}\ge s$ in problem P2-2, which is approximately convex to P2-3.

$$\text{P}2-3: \underset{\widetilde{{\varvec{w}}} ,\mathit{ s}}{\text{max}} s$$

$$\text{s}.\text{t}. {\widetilde{{\varvec{w}}}}^{H}{\left({{\varvec{h}}}_{d}^{H}+{{\varvec{f}}}_{d}^{H}\boldsymbol{\Phi }{\varvec{H}}\right)}^{H}\left({{\varvec{h}}}_{d}^{H}+{{\varvec{f}}}_{d}^{H}\boldsymbol{\Phi }{\varvec{H}}\right)\widetilde{{\varvec{w}}}\le \widetilde{r}$$

$${\widetilde{{\varvec{w}}}}^{H}\widetilde{{\varvec{w}}}\le {P}_{\text{max}}$$

$$2{\text{Re}}\left\{ {\widetilde{{\varvec{w}}}^{{(t)^{H} }} 2\widetilde{{\varvec{w}}}} \right\} \ge s + \widetilde{{\varvec{w}}}^{{(t)^{H} }} 2\widetilde{{\varvec{w}}}^{\left( t \right)}$$

Problem P2-3 is convex, which can be directly solved by using convex optimization toolbox CVX, and we obtain the optimal $\widetilde{{\varvec{w}}}$ denoted as ${\widetilde{{\varvec{w}}}}_{opt}$. Then the maximum transmit power that is practically feasible in problem P2 is ${\widetilde{P}}_{\text{max}}={\Vert {\widetilde{{\varvec{w}}}}_{\text{opt}}\Vert }^{2}$. Given the practically feasible maximum transmit power ${\widetilde{P}}_{\text{max}}$ of the base station, the optimal TPV ${\varvec{w}}$ can be found by using the method of generalized Rayleigh entropy.

Denote ${\varvec{w}}=\widehat{{\varvec{w}}}\sqrt{{\widetilde{P}}_{\text{max}}}$, where $\widehat{{\varvec{w}}}$ is the normalized vector, satisfying ${\widehat{{\varvec{w}}}}^{H}\widehat{{\varvec{w}}}=1$. Substituting ${\varvec{w}}$ into the expressions of ${C}_{c}$ and ${C}_{e}$, we have

$$\begin{aligned} 1 + {\text{SINR}}_{c} & = 1 + \frac{{\left| {\left( {{\varvec{h}}_{c}^{H} + {\varvec{f}}_{c}^{H} \user2{\Phi H}} \right){\varvec{w}}} \right|^{2} }}{{\left| {g_{c} + {\varvec{f}}_{c}^{H} \user2{\Phi g}} \right|^{2} + \sigma_{c}^{2} }} \\ & \quad = \hat{\user2{w}}^{H} \hat{\user2{w}} + \frac{{P_{max} }}{{\left| {g_{c} + {\varvec{f}}_{c}^{H} \user2{\Phi g}} \right|^{2} + \sigma_{c}^{2} }}\hat{\user2{w}}^{H} \left( {{\varvec{h}}_{c}^{H} + {\varvec{f}}_{c}^{H} \user2{\Phi H}} \right)^{H} \left( {{\varvec{h}}_{c}^{H} + {\varvec{f}}_{c}^{H} \user2{\Phi H}} \right)\hat{\user2{w}} \\ & \quad = \hat{\user2{w}}^{H} {\varvec{X}}_{c} \hat{\user2{w}} \\ \end{aligned}$$

and

$$\begin{aligned} 1 + {\text{SINR}}_{e} & = 1 + \frac{{\left| {\left( {{\varvec{h}}_{e}^{H} + {\varvec{f}}_{e}^{H} \user2{\Phi H}} \right){\varvec{w}}} \right|^{2} }}{{\left| {g_{e} + {\varvec{f}}_{e}^{H} \user2{\Phi g}} \right|^{2} + \sigma_{e}^{2} }} \\ & \quad = \hat{\user2{w}}^{H} \hat{\user2{w}} + \frac{{P_{max} }}{{\left| {g_{e} + {\varvec{f}}_{e}^{H} \user2{\Phi g}} \right|^{2} + \sigma_{e}^{2} }}\hat{\user2{w}}^{H} \left( {{\varvec{h}}_{e}^{H} + {\varvec{f}}_{e}^{H} \user2{\Phi H}} \right)^{H} \left( {{\varvec{h}}_{e}^{H} + {\varvec{f}}_{e}^{H} \user2{\Phi H}} \right)\hat{\user2{w}} \\ & \quad = \hat{\user2{w}}^{H} {\varvec{X}}_{e} \hat{\user2{w}} \\ \end{aligned}$$

where ${{\varvec{X}}}_{c}={{\varvec{I}}}_{M}+\frac{{P}_{\text{max}}}{{\left|{g}_{c}+{{\varvec{f}}}_{c}^{H}\boldsymbol{\Phi }{\varvec{g}}\right|}^{2}+{\sigma }_{c}^{2}}{\left({{\varvec{h}}}_{c}^{H}+{{\varvec{f}}}_{c}^{H}\boldsymbol{\Phi }{\varvec{H}}\right)}^{H}\left({{\varvec{h}}}_{c}^{H}+{{\varvec{f}}}_{c}^{H}\boldsymbol{\Phi }{\varvec{H}}\right)$ and ${{\varvec{X}}}_{e}={{\varvec{I}}}_{M}+\frac{{P}_{\text{max}}}{{\left|{g}_{e}+{{\varvec{f}}}_{e}^{H}\boldsymbol{\Phi }{\varvec{g}}\right|}^{2}+{\sigma }_{e}^{2}}{\left({{\varvec{h}}}_{e}^{H}+{{\varvec{f}}}_{e}^{H}\boldsymbol{\Phi }{\varvec{H}}\right)}^{H}\left({{\varvec{h}}}_{e}^{H}+{{\varvec{f}}}_{e}^{H}\boldsymbol{\Phi }{\varvec{H}}\right)$.

Based on the above discussion, problem P2 can be reformulated as unconstrained problems P2-4.

$$\text{P}2-4: \underset{{\Vert \widehat{{\varvec{w}}}\Vert }^{2}=1}{\text{max}} \frac{{\widehat{{\varvec{w}}}}^{H}{{\varvec{X}}}_{c}\widehat{{\varvec{w}}}}{{\widehat{{\varvec{w}}}}^{H}{{\varvec{X}}}_{e}\widehat{{\varvec{w}}}}$$

This problem conforms to the generalized Rayleigh entropy solution form, and the optimal $\widehat{{\varvec{w}}}$ can be obtained as ${\widehat{{\varvec{w}}}}_{\text{opt}}={\lambda }_{\text{max}}({{\varvec{X}}}_{c}{{\varvec{X}}}_{e}^{-1})$, where ${\lambda }_{\text{max}}({{\varvec{X}}}_{c}{{\varvec{X}}}_{e}^{-1})$ is the eigenvector corresponding to the largest eigenvalue of ${{\varvec{X}}}_{c}{{\varvec{X}}}_{e}^{-1}$. Finally, the optimal TPV ${\varvec{w}}$ can be given by ${{\varvec{w}}}_{\text{opt}}=\sqrt{{\widetilde{P}}_{max}}{\widehat{{\varvec{w}}}}_{opt}$.

4.2 Find the optimal RCM

This section finds the solution of RCM $\boldsymbol{\Phi }$ under the premise of fixing the TPV ${\varvec{w}}$. The original problem P1 can be reformulated as P3 with the only variable $\boldsymbol{\Phi }$ to be optimized.

$$\text{P}3: \underset{\boldsymbol{\Phi }}{\text{max}} {C}_{c}-{C}_{e}$$

$$\text{s}.\text{t}. {\text{SINR}}_{d}\ge r$$

$$\left|{\phi }_{n}\right|=1, n=\text{1,2},\cdots ,N$$

Problem P3 is still a non-convex optimization problem. Firstly, the object and constraint are simplified and replaced, and then the non-convex problem is transformed into a convex optimization problem by using convex approximation method.

Denote ${{\varvec{G}}}_{c}=\text{diag}({{\varvec{f}}}_{c}^{H})$ and ${{\varvec{G}}}_{e}=\text{diag}({{\varvec{f}}}_{e}^{H})$, then we have

$${C}_{c}={\text{log}}_{2}\left(1+\frac{{\left|\left({{\varvec{h}}}_{c}^{H}+\boldsymbol{\Phi }{{\varvec{G}}}_{c}{\varvec{H}}\right){\varvec{w}}\right|}^{2}}{{\left|{g}_{c}+\boldsymbol{\Phi }{{\varvec{G}}}_{c}{\varvec{g}}\right|}^{2}+{\sigma }_{c}^{2}}\right)$$

and

$${C}_{e}={\text{log}}_{2}\left(1+\frac{{\left|\left({{\varvec{h}}}_{e}^{H}+\boldsymbol{\Phi }{{\varvec{G}}}_{e}{\varvec{H}}\right){\varvec{w}}\right|}^{2}}{{\left|{g}_{e}+\boldsymbol{\Phi }{{\varvec{G}}}_{e}{\varvec{g}}\right|}^{2}+{\sigma }_{e}^{2}}\right)$$

Let ${\varvec{z}}=\left(\begin{array}{c}\begin{array}{c}{\phi }_{1}\\ \vdots \end{array}\\ {\phi }_{N}\\ 1\end{array}\right)$, ${\widetilde{{\varvec{G}}}}_{c}=\left(\begin{array}{c}{{\varvec{G}}}_{c}{\varvec{H}}\\ {{\varvec{h}}}_{c}^{H}\end{array}\right)$, ${\overline{{\varvec{G}}} }_{c}=\left(\begin{array}{c}{{\varvec{G}}}_{c}{\varvec{g}}\\ {g}_{c}\end{array}\right)$, ${\widetilde{{\varvec{G}}}}_{e}=\left(\begin{array}{c}{{\varvec{G}}}_{c}{\varvec{H}}\\ {{\varvec{h}}}_{e}^{H}\end{array}\right)$, ${\overline{{\varvec{G}}} }_{e}=\left(\begin{array}{c}{{\varvec{G}}}_{c}{\varvec{g}}\\ {g}_{e}\end{array}\right)$, ${\widetilde{{\varvec{G}}}}_{d}=\left(\begin{array}{c}{{\varvec{G}}}_{d}{\varvec{H}}\\ {{\varvec{h}}}_{d}^{H}\end{array}\right)$ and ${\overline{{\varvec{G}}} }_{d}=\left(\begin{array}{c}{{\varvec{G}}}_{d}{\varvec{g}}\\ {g}_{d}\end{array}\right)$, ${C}_{c}$ and ${C}_{e}$ can be rewritten as

$$\begin{aligned} C_{c} & = \log_{2} \left( {1 + \frac{{\left| {{\varvec{z}}^{H} \tilde{\user2{G}}_{c} {\varvec{w}}} \right|^{2} }}{{\left| {{\varvec{z}}^{H} \overline{\user2{G}}_{c} } \right|^{2} + \sigma_{c}^{2} }}} \right) \\ & \quad = \log_{2} \left( {\frac{{{\varvec{z}}^{H} \left( {\overline{\user2{G}}_{c} \overline{\user2{G}}_{c}^{H} + \tilde{\user2{G}}_{c} {\varvec{ww}}^{H} \tilde{\user2{G}}_{c}^{H} } \right){\varvec{z}} + \sigma_{c}^{2} }}{{{\varvec{z}}^{H} \overline{\user2{G}}_{c} \overline{\user2{G}}_{c}^{H} {\varvec{z}} + \sigma_{c}^{2} }}} \right) \\ \end{aligned}$$

and

$$\begin{aligned} C_{e} & = \log_{2} \left( {1 + \frac{{\left| {{\varvec{z}}^{H} \tilde{\user2{G}}_{e} {\varvec{w}}} \right|^{2} }}{{\left| {{\varvec{z}}^{H} \overline{\user2{G}}_{e} } \right|^{2} + \sigma_{c}^{2} }}} \right) \\ & \quad = \log_{2} \left( {\frac{{{\varvec{z}}^{H} \left( {\overline{\user2{G}}_{e} \overline{\user2{G}}_{e}^{H} + \tilde{\user2{G}}_{e} {\varvec{ww}}^{H} \tilde{\user2{G}}_{e}^{H} } \right){\varvec{z}} + \sigma_{e}^{2} }}{{{\varvec{z}}^{H} \overline{\user2{G}}_{e} \overline{\user2{G}}_{e}^{H} {\varvec{z}} + \sigma_{e}^{2} }}} \right) \\ \end{aligned}$$

Therefore, the objective function of problem P3 is equivalent to

$$\begin{gathered} \log_{2} \left( {\frac{{{\varvec{z}}^{H} \left( {\overline{\user2{G}}_{c} \overline{\user2{G}}_{c}^{H} + \tilde{\user2{G}}_{c} {\varvec{ww}}^{H} \tilde{\user2{G}}_{c}^{H} } \right){\varvec{z}} + \sigma_{c}^{2} }}{{{\varvec{z}}^{H} \overline{\user2{G}}_{c} \overline{\user2{G}}_{c}^{H} {\varvec{z}} + \sigma_{c}^{2} }}} \right) \hfill \\ \quad - \log_{2} \left( {\frac{{{\varvec{z}}^{H} \left( {\overline{\user2{G}}_{e} \overline{\user2{G}}_{e}^{H} + \tilde{\user2{G}}_{e} {\varvec{ww}}^{H} \tilde{\user2{G}}_{e}^{H} } \right){\varvec{z}} + \sigma_{e}^{2} }}{{{\varvec{z}}^{H} \overline{\user2{G}}_{e} \overline{\user2{G}}_{e}^{H} {\varvec{z}} + \sigma_{e}^{2} }}} \right) \hfill \\ \end{gathered}$$

(10)

The constraint ${\text{SINR}}_{d}\ge r$ can be reformulated as

$${\left|{{\varvec{z}}}^{H}{\overline{{\varvec{G}}} }_{d}\right|}^{2}-r{\sigma }_{d}^{2}\ge r{\left|{{\varvec{z}}}^{H}{\widetilde{{\varvec{G}}}}_{d}{\varvec{w}}\right|}^{2}$$

which can be further equivalent to

$${{\varvec{z}}}^{H}{\overline{{\varvec{G}}} }_{d}{\overline{{\varvec{G}}} }_{d}^{H}{\varvec{z}}\ge r{\sigma }_{d}^{2}+r{{\varvec{z}}}^{H}{\widetilde{{\varvec{G}}}}_{d}{\varvec{w}}{{\varvec{w}}}^{H}{\widetilde{{\varvec{G}}}}_{d}^{H}{\varvec{z}}$$

(11)

However, the objective function (10) is still non-concave; introduce slack variables $\widetilde{{\varvec{p}}}=\left({p}_{1}, {p}_{2},{p}_{3},{p}_{4}\right)$ and set

$$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\log_{2} \left( {{\varvec{z}}^{H} \left( {\overline{\user2{G}}_{c} \overline{\user2{G}}_{c}^{H} + \tilde{\user2{G}}_{c} {\varvec{ww}}^{H} \tilde{\user2{G}}_{c}^{H} } \right){\varvec{z}} + \sigma_{c}^{2} } \right) \ge p_{1} } & {\quad \left( {12{\text{a}}} \right)} \\ {\log_{2} \left( {{\varvec{z}}^{H} \overline{\user2{G}}_{c} \overline{\user2{G}}_{c}^{H} {\varvec{z}} + \sigma_{c}^{2} } \right) \le p_{2} } & {\quad \left( {12{\text{b}}} \right)} \\ \end{array} } \\ {\begin{array}{*{20}c} {\log_{2} \left( {{\varvec{z}}^{H} \left( {\overline{\user2{G}}_{e} \overline{\user2{G}}_{e}^{H} + \tilde{\user2{G}}_{e} {\varvec{ww}}^{H} \tilde{\user2{G}}_{e}^{H} } \right){\varvec{z}} + \sigma_{e}^{2} } \right) \le p_{3} } & {\quad \left( {12{\text{c}}} \right)} \\ {\log_{2} \left( {{\varvec{z}}^{H} \overline{\user2{G}}_{e} \overline{\user2{G}}_{e}^{H} {\varvec{z}} + \sigma_{e}^{2} } \right) \ge p_{4} } & {\quad \left( {12{\text{d}}} \right)} \\ \end{array} } \\ \end{array} } \right.$$

such that ${C}_{c}\ge {p}_{1}-{p}_{2}$ and ${C}_{e}\le {p}_{3}-{p}_{4}$.

Thus, problem P3 can be transformed into P3-1

$$\text{P}3-1: \underset{{\varvec{z}},\boldsymbol{ }\widetilde{{\varvec{p}}}\boldsymbol{ }}{\text{max}} {p}_{1}-{p}_{2}-{p}_{3}+{p}_{4}$$

s.t. (11), (12)

$$\left|{z}_{n}\right|=1, n=\text{1,2},\cdots ,N$$

In P3-1, the SINR constraint (11) and RIS phase shift constraint are still non-convex. Let $\widetilde{{\varvec{Z}}}={\varvec{z}}{{\varvec{z}}}^{H}$, then the constraint (11) is equivalent to

$$\text{Tr}\left\{\widetilde{{\varvec{Z}}}{\overline{{\varvec{G}}} }_{d}{\overline{{\varvec{G}}} }_{d}^{H}\right\}\ge r{\sigma }_{d}^{2}+r\text{Tr}\left\{\widetilde{{\varvec{Z}}}{\widetilde{{\varvec{G}}}}_{d}{{\varvec{w}}{{\varvec{w}}}^{H}\widetilde{{\varvec{G}}}}_{d}^{H}\right\}$$

(13)

and the constraint $\left|{z}_{n}\right|=1, n=\text{1,2},\cdots ,N$ is equivalent to

$$\left\{\widetilde{{\varvec{Z}}}\succcurlyeq 0,\text{rank}\left(\widetilde{{\varvec{Z}}}\right)=1,\widetilde{\text{diag}}\left(\widetilde{{\varvec{Z}}}\right)={{\varvec{I}}}_{N+1}\right\}$$

(14)

In order to see the convexity more clearly from (12), another slack variable ${\widetilde{{\varvec{s}}}}_{z}=\left({s}_{z,1}, {s}_{z,2},{s}_{z,3},{s}_{z,4}\right)$ is introduced, and (12) can be further given by

$$\left( {12{\text{a}}} \right)\left( {12{\text{b}}} \right) \Rightarrow \left\{ {\begin{array}{*{20}c} {\log_{2} \left( {s_{z,1} } \right) \ge p_{1} } & {\left( {15{\text{a}}} \right)} \\ {\log_{2} \left( {s_{z,2} } \right) \le p_{2} } & {\left( {15{\text{b}}} \right)} \\ {{\text{Tr}}\left\{ {\tilde{\user2{Z}}\left( {\overline{\user2{G}}_{c} \overline{\user2{G}}_{c}^{H} + \tilde{\user2{G}}_{c} {\varvec{ww}}^{H} \tilde{\user2{G}}_{c}^{H} } \right)} \right\} + \sigma_{c}^{2} \ge s_{z,1} } & {\left( {15{\text{c}}} \right)} \\ {{\text{Tr}}\left\{ {\user2{\tilde{Z}\overline{G}}_{c} \overline{\user2{G}}_{c}^{H} } \right\} + \sigma_{c}^{2} \le s_{z,2} } & {\left( {15{\text{d}}} \right)} \\ \end{array} } \right.$$

$$\left( {12{\text{c}}} \right)\left( {12{\text{d}}} \right) \Rightarrow \left\{ {\begin{array}{*{20}c} {\log_{2} \left( {s_{z,3} } \right) \le p_{3} } & {\left( {16{\text{a}}} \right)} \\ {\log_{2} \left( {s_{z,4} } \right) \ge p_{4} } & {\left( {16{\text{b}}} \right)} \\ {{\text{Tr}}\left\{ {\tilde{\user2{Z}}\left( {\overline{\user2{G}}_{e} \overline{\user2{G}}_{e}^{H} + \tilde{\user2{G}}_{e} {\varvec{ww}}^{H} \tilde{\user2{G}}_{e}^{H} } \right)} \right\} + \sigma_{e}^{2} \le s_{z,3} } & {\left( {16{\text{c}}} \right)} \\ {{\text{Tr}}\left\{ {\user2{\tilde{Z}\overline{G}}_{e} \overline{\user2{G}}_{e}^{H} } \right\} + \sigma_{e}^{2} \ge s_{z,4} } & {\left( {16{\text{d}}} \right)} \\ \end{array} } \right.$$

The constraint (11) can be replaced by (13), the constraint (12) can be replaced by (15) and (16), and the phase shift constraint can be replaced by (14). However, both the functions (15b) and (16a) are still non-convex. The first-order Taylor expansion is used again to find the lower bound such that (15b) and (16a) can be approximated to convex form, as follows

$$\begin{array}{*{20}c} {\left( {15{\text{b}}} \right) \Rightarrow \log_{2} \left( {s_{z,2}^{\left( t \right)} } \right) + \frac{{s_{z,2} - s_{z,2}^{\left( t \right)} }}{{s_{z,2}^{\left( t \right)} \ln 2}} \le p_{2} } & {\quad \left( {17{\text{a}}} \right)} \\ {\left( {16{\text{a}}} \right) \Rightarrow \log_{2} \left( {s_{z,3}^{\left( t \right)} } \right) + \frac{{s_{z,3} - s_{z,3}^{\left( t \right)} }}{{s_{z,3}^{\left( t \right)} \ln 2}} \le p_{3} } & {\quad \left( {17{\text{b}}} \right)} \\ \end{array}$$

Finally, the optimal problem P3-1 can be reformulated as P3-2

$$\text{P}3-2: \underset{\widetilde{{\varvec{z}}},\boldsymbol{ }\widetilde{{\varvec{p}},}{\widetilde{{\varvec{s}}}}_{z}\boldsymbol{ }}{\text{max}} {p}_{1}-{p}_{2}-{p}_{3}+{p}_{4}$$

s.t. (13), $\left\{\widetilde{{\varvec{Z}}}\succcurlyeq 0,\text{rank}\left(\widetilde{{\varvec{Z}}}\right)=1,\widetilde{\text{diag}}\left(\widetilde{{\varvec{Z}}}\right)={{\varvec{I}}}_{N+1}\right\}$,

(15a), (15c), (15d), (16b), (16c), (16d), (17)

We can solve problem P3-2 without considering the constraint $\text{rank}\left(\widetilde{{\varvec{Z}}}\right)=1$, that is, we can relax P3-2 to an SDP problem. The relaxed SDP problem is a convex optimization problem, which can be solved by using CVX. However, the optimal solution ${\widetilde{{\varvec{Z}}}}_{\text{opt}}$ for the SDP problem does not necessarily satisfy $\text{rank}\left({\widetilde{{\varvec{Z}}}}_{\text{opt}}\right)=1$, so the Gaussian randomization method is used to find the optimal ${\widetilde{{\varvec{Z}}}}_{\text{opt}}$ satisfying $\text{rank}\left(\widetilde{{\varvec{Z}}}\right)=1$. The process of Gaussian randomization to determine the optimal ${\varvec{z}}$ is listed in Algorithm 1. Given the optimal ${\varvec{z}}$, we can obtain the optimal ${\varvec{\phi}}$.

Algorithm 1

Gaussian randomization

4.3 The overall algorithm and complexity analysis

Based on the above two sub-problems in Sect. 3.1 and Sect. 3.2, the overall algorithm description of problem P1 is given in Algorithm 2. It can be seen from the algorithm that the two sub-problems of optimizing TPV and RCM mainly involve SOC constraints and matrix inversion operations; hence, the complexity of the algorithm is $\mathcal{O}\left({M}^{3}\right)+\mathcal{O}({N}^{3})$.

Algorithm 2

Joint TPV and RCM optimization algorithm

5 Simulation results and discussion

In order to verify the effectiveness of the proposed algorithm, a two-lane IoV simulation scenario is set up, and the coordinate settings of each part are shown in Fig. 2, in which the positions of B and RIS are fixed in $(0, 0)$ and $(0, 8)$, and the positions of vehicles are random and change with the driving of vehicles. The average speed of vehicles is $v$, and the average interval of vehicles is 2.5 $v$ [9]. Unless some parameters are specified in simulation, other parameters are set as shown in Table 2. According to the current allocation of the communication frequency 6 GHz for the IoV and the typical range of vehicle speeds [40 km/h, 120 km/h], the Doppler shift ${f}_{d}$ is in the range of [218.5 Hz, 655.6 Hz]. From the simulation parameters of Table 2, we can see that the system spectral bandwidth is much larger than the Doppler bandwidth, i.e., $B\gg 2{f}_{d}$. Therefore, we do not consider fast fading caused by Doppler shift, and only consider frequency selective fading in our simulations. The channels associated with the RIS, including the channels ${\varvec{H}}$, ${\varvec{g}}$, ${{\varvec{f}}}_{c}$ and ${{\varvec{f}}}_{e}$, are assumed to be Rician channels due to the beam-directional capability of the RIS. The other channels are assumed to be Rayleigh channels. The Rician channel is described by the following equation.

$${\varvec{f}}=\sqrt{\rho {d}^{-\alpha }}\left(\sqrt{\frac{\kappa }{1+\kappa }}{{\varvec{f}}}^{\text{LoS}}+\sqrt{\frac{1}{1+\kappa }}{{\varvec{f}}}^{\text{NLoS}}\right)$$

where $d$ is the distance between transceiver, $\rho$ is the path loss at a reference distance of 1 m, $\alpha$ is channel path loss factor, $\kappa$ is Rician fading factor, ${{\varvec{f}}}^{\text{NLoS}}$ is a non-line-of-sight random scattering path, characterized by a standard complex Gaussian distribution, and ${{\varvec{f}}}^{\text{LoS}}$ is the line-of-sight path, which can be characterized by antenna or phase shift array reactions at both transmitter and receiver. The Rayleigh channel can be characterized by the following equation.

$${\varvec{g}}=\sqrt{\rho {d}^{-\alpha }}\widehat{{\varvec{g}}}$$

where $\widehat{{\varvec{g}}}$ is random scattering path, characterized by a standard complex Gaussian distribution.

Fig. 2

The 2D simulation scenario. A two-lane IoV simulation scenario, in which the positions of B and RIS are fixed in $(0, 0)$ and $(0, 8)$, and the positions of vehicles are random and change with the driving of vehicles. The average speed of vehicles is $v$

Table 2 Simulation parameters

Figure 3 shows the convergence results of the proposed algorithm. When the number of RIS reflection units is different, such as $N=4, 16, 64$, the proposed algorithm can obtain the stable optimal solution within about 10 iterations, which verifies that the proposed algorithm in this paper has good convergence.

Fig. 3

Algorithm convergence. The figure shows the convergence results of the proposed algorithm. When the number of RIS reflection units is different, such as $N=4, 16, 64$, the proposed algorithm can obtain the stable optimal solution within about 10 iterations, which verifies that the proposed algorithm in this paper has good convergence

Figure 4 shows the effect of the number of RIS reflection units on the PLSTR. With the increase of the number of RIS reflection units, the PLSTR gradually increases. For example, when the number of transmitting antennas at base station is 2 and the number of RIS reflection units increases from 16 to 100, the PLSTR can be doubled. With the increase in number of the RIS reflection units, the reflection shifts can be adjusted more flexibly, so as to achieve stronger reflection beamforming effect and improve the system security transmission rate. Moreover, it can be seen from the three curves in the figure that the increase of the number of transmitting antennas at base station can further improving the level of PLSTR; this is because the multiplexing and diversity gains brought by more multiple antennas are more obvious.

Fig. 4

Effect of the number of RIS reflection units on the PLSTR. The figure shows the effect of the number of RIS reflection units on the PLSTR. With the increase of the number of RIS reflection units, the PLSTR gradually increases. Moreover, it can be seen that the increase of the number of transmitting antennas at base station can further improve the level of PLSTR

Figure 5 shows the performance comparison between the proposed algorithm and other algorithms. The “Rand RIS Phase” algorithm means that the RCM at RIS is randomly given, and the TPV is obtained by the proposed method in this paper. The “Channel Matching Algorithm” means that the TPV adopts the ideal form and is designed by using the singular value decomposition of the channel gain matrix, and the RCM of the RIS is obtained by the optimization method proposed in this paper. The “SOCP + SDP” algorithm means that the solutions of the TPV and the RCM of RIS are obtained by directly transforming the prime optimization problem into a classical second-order cone programming (SOCP) sub-problem and a semi-definite programming (SDP) sub-problem, respectively, instead of using the method of generalized Rayleigh entropy proposed in this paper. Through comparison, it is found that the system performance of the proposed algorithm is similar to “SOCP + SDP” algorithm, and better than “Rand RIS Phase” algorithm and “Channel Matching” algorithm. However, the complexity of the proposed algorithm is $\mathcal{O}\left({M}^{3}\right)+\mathcal{O}({N}^{3})$, which is slightly lower than $\mathcal{O}\left({M}^{3.5}\right)+\mathcal{O}({N}^{3})$ of “SOCP + SDP” algorithm due to the analytical form of generalized Rayleigh entropy.

Fig. 5

Comparison between different algorithms. The figure shows the performance comparison between the proposed algorithm and other algorithms. The proposed algorithm is similar to “SOCP + SDP” algorithm, and better than “Rand RIS Phase” algorithm and “Channel Matching” algorithm

Figure 6 further shows the effect of the maximum transmit power of base station and the path loss factor on the PLSTR. The larger the maximum transmission power of base station is, the higher the PLSTR is, because with the increase of the transmission power of the base station, the communication rate of V2I becomes higher, and meanwhile the RIS adjusts the phase to suppress the eavesdropping path, so that the eavesdropping received signal is suppressed and the security transmission signal is enhanced. The trend comparison of the three solid lines in the figure also shows that under the same number of RIS reflection units, the larger the path loss factor ${\alpha }_{B2E}$ between B and E is, the larger the PLSTR of V2I is, because the increase of the path loss factor between B and E will reduce the eavesdropping rate, thereby increasing the security transmission rate.

Fig. 6

Effect of the maximum transmit power of base station on the PLSTR. The figure shows the effect of the maximum transmit power of base station and the path loss factor on the PLSTR. The larger the maximum transmission power of base station is, the higher the PLSTR is. Also it shows that the larger the path loss factor ${\alpha }_{B2E}$ between B and E is, the larger the PLSTR of V2I is

Figure 7 shows the effect of the average vehicle speed on the PLSTR. Since the Doppler shift does not cause fast fading, the change of vehicle speed mainly affects the spacing between vehicles. According to our assumption, the larger the vehicle speed, the larger the vehicle spacing, the higher the path loss between V2V and V2I, and the lower the interference between each other. At the same time, the received signal power of the eavesdropping vehicle is also weakened, and combined with the control of RIS, the PLSTR of V2I gradually increases. However, this increasing trend will gradually level off with the increase of vehicle speed. This is because the PLSTR is capped and it will not be higher than the information transmission rate of V2I. In addition, it can be found from the figure that the path loss factor between the base station and V2I has a significant impact on the performance. When the factor is reduced from 3 to 2, the PLSTR increases by 3 dB.

Fig. 7

Effect of the average vehicle speed on the PLSTR

6 Conclusion

In this paper, the communication scenario where V2I and V2V share the spectrum at the same time is modeled. By jointly optimizing the base station transmission precoding vector and RIS reflection coefficient matrix, an alternating optimization algorithm based on generalized Rayleigh entropy is proposed, which can improve the V2I physical layer security transmission rate in the coexistence of V2I and V2V network. The proposed algorithm converges faster and has better performance than the classical algorithm and SOCP plus SDP algorithm. In the case of increasing the number of RIS reflection units and the number of base station transmitter antennas, the V2I physical layer security transmission rate can be greatly improved.