Low-Complexity Sum-Rate Maximization for Multi-IRS-Assisted V2I Systems

Abstract

Intelligent reflecting surface (IRS) has emerged as a promising solution to establish propagation paths in non-line-of-sight (NLoS) scenarios, effectively mitigating blockage challenges in direct vehicle-to-infrastructure (V2I) links. This study investigates a time-varying multi-IRS-assisted multiple-input multiple-output (MIMO) communication system, aiming to maximize the system sum rate through the joint optimization of base station (BS) precoding and IRS phase configurations. The formulated problem exhibits inherent non-convexity and time-varying characteristics, posing significant optimization challenges. To address these, we propose a low-complexity dimension-wise sine maximization (DSM) algorithm, grounded in the sum path gain maximization (SPGM) criterion, to efficiently optimize the IRS phase shift matrix. Concurrently, the water-filling (WF) algorithm is employed for BS precoding design. Simulation results demonstrate that compared with traditional methods, the proposed DSM algorithm achieves a 14.9% increase in sum rate, while exhibiting lower complexity and faster convergence. Furthermore, the proposed multi-IRS design yields an 8.7% performance gain over the single-IRS design.

1. Introduction

Vehicle-to-Infrastructure (V2I) communication, a key 6G application scenario, demonstrates critical value in fields like vehicular networks [1]. With the proliferation of services such as autonomous driving and real-time traffic monitoring, V2I systems face stringent requirements for high throughput and low latency. However, factors including building obstructions in urban environments and high-speed vehicle mobility lead to non-line-of-sight (NLoS) propagation and Doppler effects, severely compromising the reliability and spectral efficiency of conventional V2I links [2]. To enhance V2I communication quality, intelligent reflecting surface (IRS) emerges as a promising solution [3]. More specifically, an IRS comprises a digitally controlled metasurface with numerous passive reflecting elements. Each element independently adjusts the phase shifts of incident signals in real time [4]. This enables the IRS to establish effective virtual line-of-sight (LoS) links and perform passive beamforming, thereby actively shaping the wireless propagation environment to achieve efficient, secure, and reliable wireless networks [5]. Furthermore, IRS technology effectively compensates for Doppler-induced signal distortions [6].

1.1. Related Work

Recent years have seen significant advancements in the optimization of wireless networks enabled by IRS technology. In research focused on single-IRS scenarios, the authors of [7,8] investigated the system power minimization problem for multi-user multiple-input single-output (MU-MISO) downlinks, providing crucial solutions for enhancing energy efficiency. The research in [9] explored IRS-assisted UAV communications, offering a novel perspective on IRS applications in dynamic environments while highlighting the potential of IRS in enhancing the reliability of dynamic links. For the more complex IRS-assisted multiple-input multiple-output (MIMO) systems, system capacity maximization emerges as a core challenge. In [10,11,12,13], the authors effectively addressed this issue through the joint optimization of the IRS phase shift matrix and the transmit covariance matrix, significantly improving spectral efficiency. Given the inherently high computational complexity of the capacity maximization problem itself, the authors of [14,15] innovatively proposed the sum path gain maximization (SPGM) criterion. This criterion significantly reduces algorithmic complexity while ensuring near-optimal performance by optimizing the coherent superposition of signal paths via IRS elements. In addition to data transmission, IRS has also demonstrated potential in the field of wireless power transmission [16,17]. To further extend coverage, multi-IRS collaborative assistance has become a research hotspot. The core concept of [18,19,20] lies in leveraging the short-range coverage characteristics of a single IRS to achieve long-distance, high-reliability signal transmission through the distributed deployment and cascaded reflections of multiple intelligent reflecting surfaces (IRSs). Research in [21] further confirms that distributed IRS deployment can substantially enhance the overall performance of wireless systems, offering a new enabling technological pathway for future ultra-dense or wide-area coverage networks.

In the challenging scenario of V2I communication, IRS technology also demonstrates significant application potential. To accurately characterize channel properties, the authors of [22,23] proposed a novel non-stationary three-dimensional (3D) wideband channel model for double-IRS assisted MIMO systems. Given that high-speed vehicle movement introduces substantial Doppler shift effects, Lian et al. in [24] derived and analyzed the spatial–temporal correlation functions and channel capacity. Their research revealed that adjusting the IRS phase shifts can mitigate multipath and Doppler effects. Complementing this, the authors of [25] investigated the functional mechanism of IRS in compensating for Doppler effects in mobile communication systems. To maximize system throughput, the authors of [26,27] employed deep reinforcement learning (DRL) and unsupervised learning methods, respectively, to collaboratively optimize the base station (BS) beamforming and IRS reflection coefficients. More specifically, Ma et al. in [28] derived expressions for the received signal power and the upper bound of the ergodic sum capacity in the V2I system, achieved by comprehensively considering the horizontal and vertical rotation angles of the IRS elements and utilizing carefully designed reflection phases. Addressing the stringent reliability requirements of vehicular networks, the authors of [29] focused on optimizing the IRS reflection coefficients to satisfy specific communication reliability metrics. Furthermore, the authors of [30,31] proposed a IRS-assisted hybrid access scheme designed to balance the demands for high reliability and high throughput. Additionally, a thorough analysis has been conducted on IRS-assisted V2I systems’ performance parameters, including outage probability [32] and energy efficiency [33]. These analyses provide vital assessment benchmarks for evaluating the practical deployment efficacy of IRS in vehicular networks.

1.2. Motivations

The research efforts regarding the optimization of the sum rate for this topic are summarized in

Table 1. Relevant works on the topic.

Literature	System Mode	Method	Consider the Doppler Effects?	Complexity
[4]	Single-input–single-output (SISO)	Successive convex approximation (SCA)	No	Modest
[8]	MU-MISO	Alternating optimization (AO)	No	High
[10,12]	MIMO	AO, Projected gradient	No	High
[26]	MIMO	DRL	Yes	High
[27]	MU-MISO	Unsupervised learning	Yes	Modest

. As shown in Table 1, due to the high complexity of the system architecture, there is a lack of research on capacity optimization for IRS-assisted V2I systems. Although existing methods are effective, they are mainly oriented to low-mobility users with slow-fading channel scenarios, ignoring the effect of Doppler effect on the system and high computational complexity, which makes them difficult to apply to vehicular networking scenarios. This compels us to suggest a method with reduced computational complexity. Based on the channel state information (CSI), the proposed algorithm can be adapted to the real-time changes of the channel and can effectively improve the upper capacity limit of MIMO systems.

1.3. Contributions

In this paper, we study the influence of the Doppler effect on channel modeling for multi-IRS-assisted V2I communication systems and develop a SPGM-based algorithm for MIMO capacity maximization. The principal contributions are delineated as follows:

• We analyze the Doppler shift of the LoS component and the Doppler expansion of the non-line-of-sight (NLoS) component of the association link of a vehicle in a moving state, and the time-varying communication model for multi-IRS-assisted V2I is established. The V2I capacity is maximized by the joint optimization of base station (BS) active precoding and IRS passive phase shifting.
• The original problem is decomposed into two subproblems: precoding and IRS phase-shift optimization. To address the nonconvexity of the IRS phase-shift subproblem, the problem is simplified based on the sum path gain maximization (SPGM) criterion proposed in [14] to indirectly optimize the sum path gain, and a new algorithm named dimension-wise sine maximization (DSM) is proposed to directly tackle the problem with lower complexity. Then, the precoding of the BS is derived from the obtained IRS phase shift matrix using the water-filling (WF) algorithm.

The remainder of the paper is structured as follows. Section 2 describes the multi-IRS-assisted V2I communication system model and constructs the sum-rate maximization problem based on this model. The DSM algorithm used for optimal design is presented in Section 3. Numerical results are given in Section 4. Finally, Section 5 summarizes the conclusions of this paper.

2. System Model

As shown in Figure 1, we consider a multi-IRS assisted 3D MIMO mobile communication scenario comprising a static BS, a mobile receiver (MR) moving with velocity $v_R$ and azimuth angle ${\gamma }_R$, and $N_s$ IRS units. Each IRS is equipped with a uniform planar array (UPA) consisting of $N_{IRS}=N_x\times N_y$ reflecting elements (for mathematical simplicity, we assume all IRSs have the same number of elements, with the total number of reflecting elements across all IRSs being $N=N_s\times N_{IRS}$. However, the proposed system model and algorithm remain applicable to IRSs with heterogeneous element counts). Both the BS and MR are equipped with omnidirectional uniform linear arrays (ULAs) with $N_T$ and $N_R$ antennas, respectively. The adjacent antenna spacing at the BS and MR, as well as the inter-element spacing at the IRSs, is set to $\delta$.

We focus on a typical transmission frame comprising T time slots in this paper, denoted as $t\in \mathcal{T}\triangleq \{1,\dots ,T\}$, where each time slot has a duration $T_b$. To ensure the signal propagation characteristics remain quasi-static within each time slot, $T_b$ is set to a value smaller than the minimum coherence time across all participating communication links [34]. This assumption holds when the MR’s velocity and direction remain constant. During time slot t, the BS transmits $S\left(t\right)$ parallel data streams to the MR, represented as $\mathbf{s}\left(t\right)\in {\mathbb{C}}^{S\left(t\right)\times 1}$, satisfying $\mathbb{E}\left[\mathbf{s}\left(t\right)\mathbf{s}{\left(t\right)}^H\right]={\mathbf{I}}_{S\left(t\right)}$, where $S\left(t\right)\le N_T$. The signals are broadcast to all IRSs and the MR via linear precoding $\mathbf{F}\left(t\right)\in {\mathbb{C}}^{N_T\times S\left(t\right)}$.

Define the time-varying channel from the BS to the mobile receiver (MR) as $\mathbf{D}\left(t\right)\in {\mathbb{C}}^{N_R\times N_T}$. Let ${\mathbf{M}}_i\in {\mathbb{C}}^{N_{IRS}\times N_T}$ represent the quasi-static channel from the BS to the i-th IRS ($i=1,\dots ,N_s$), and ${\mathbf{G}}_i\left(t\right)\in {\mathbb{C}}^{N_R\times N_{IRS}}$ characterize the dynamic IRS-to-MR link for the i-th IRS. The reflection matrix of the i-th IRS at time slot t is defined as ${\mathbf{\Theta }}_i\left(t\right)=\mathrm{diag}[e^{j{\theta }_1^i\left(t\right)},e^{j{\theta }_2^i\left(t\right)},\dots ,e^{j{\theta }_{N_{IRS}}^i\left(t\right)}]\in {\mathbb{C}}^{N_{IRS}\times N_{IRS}}$ and ${\theta }_n^i\left(t\right)\in [0,2\pi )$ represents the phase shift of the n-th reflecting element at the i-th IRS. Due to significant path loss, only single-bounce reflections via IRSs are considered. Additionally, the Rician fading model is adopted to characterize the characteristics of all channels:

$$\mathbf{H}\left(t\right)=\sqrt{\rho d^{-\alpha }}\left(\sqrt{{\displaystyle \frac{\kappa }{1+\kappa }}}{\mathbf{H}}^{\mathrm{LoS}}\left(t\right)+\sqrt{{\displaystyle \frac{1}{1+\kappa }}}{\mathbf{H}}^{\mathrm{NLoS}}\left(t\right)\right),$$

(1)

where $\rho$ is the path loss at the reference distance $d_0=1\mathrm{m}$, d is the transmitter–receiver distance, and $\alpha$ is the path loss exponent of the communication link.

For the NLoS channel component ${\mathbf{H}}^{\mathrm{NLoS}}\left(t\right)$, since ${\mathbf{G}}_i^{\mathrm{NLoS}}\left(t\right)$ and ${\mathbf{D}}^{\mathrm{NLoS}}\left(t\right)$ are time-varying due to MR mobility, they are modeled as Rayleigh-distributed with Jakes power spectral density. In contrast, ${\mathbf{M}}_i^{\mathrm{NLoS}}$ remains static over time, with its elements independently sampled from a circularly symmetric complex normal distribution $\mathcal{CN}(0,1)$.

For the LoS channel component ${\mathbf{H}}^{\mathrm{LoS}}\left(t\right)$, we incorporate Doppler effects due to MR mobility [35]. Since the BS and MR are equipped with ULAs and the IRSs are equipped with UPAs, the LoS channel components are expressed as follows:

$$\begin{array}{cc}\hfill {\mathbf{D}}^{\mathrm{LoS}}\left(t\right)& ={\mathbf{a}}_R\left({\vartheta }_{R,T}\right){\mathbf{a}}_T^H\left({\vartheta }_{T,R}\right)e^{-j{\displaystyle \frac{2\pi }{\lambda }}{v}_R{T}_{b}tcos({\vartheta }_{R,T}-{\gamma }_R)cos{\varphi }_{R,T}}\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill {\mathbf{G}}_i^{\mathrm{LoS}}\left(t\right)& ={\mathbf{a}}_R\left({\vartheta }_{R,i}\right){\mathbf{a}}_{i,t}^H({\vartheta }_{i,t},{\varphi }_{i,t})e^{-j{\displaystyle \frac{2\pi }{\lambda }}{v}_R{T}_{b}tcos({\vartheta }_{R,i}-{\gamma }_R)cos{\varphi }_{R,i}}\hfill \end{array}$$

(2b)

$$\begin{array}{cc}\hfill {\mathbf{M}}_i^{\mathrm{LoS}}& ={\mathbf{a}}_{i,r}({\vartheta }_{i,r},{\varphi }_{i,r}){\mathbf{a}}_T^H\left({\vartheta }_{T,i}\right)\hfill \end{array}$$

(2c)

where ${\mathbf{a}}_R\left({\vartheta }_{R,T/i}\right)\in {\mathbb{C}}^{N_R\times 1}$ and ${\mathbf{a}}_T\left({\vartheta }_{T,R/i}\right)\in {\mathbb{C}}^{N_T\times 1}$ represent the array steering vector of the MR and BS antennas and ${\mathbf{a}}_{i,r/t}({\vartheta }_{i,r/t},{\varphi }_{i,r/t})\in {\mathbb{C}}^{N_{\mathrm{IRS}}\times 1}$ represents the array steering vector of the i-th IRS receiving/transmitting direction. In addition, ${\vartheta }_{R,T/i}\in [0,\pi )$, ${\varphi }_{R,T/i}\in [0,{\displaystyle \frac{\pi }{2}})$ are the azimuth and elevation angles of the direction of arrival (DOA) from the BS/ i-th IRS to the MR; ${\vartheta }_{T,R/i}\in [0,\pi )$, ${\varphi }_{T,R/i}\in [0,{\displaystyle \frac{\pi }{2}})$ are the azimuth and elevation angles of the direction of departure (DOD) from the BS to the MR/ i-th IRS; and ${\vartheta }_{i,t/r}\in [0,\pi )$, ${\varphi }_{i,t/r}\in [0,{\displaystyle \frac{\pi }{2}})$ are the azimuth and elevation angles of the DOD/DOA from the i-th IRS, respectively. The carrier wavelength $\lambda$ governs the phase coherence across channels, while the exponential terms in (2a) and (2b) model Doppler shifts induced by MR mobility with velocity $v_R$ and azimuth angle ${\gamma }_R$. All angles can be calculated based on the real-time position information of the vehicle. The array steering vector $\mathbf{a}\left(\theta \right)$ for a ULA is generally expressed as follows:

$$\mathbf{a}\left(\theta \right)={[1e^{j{\displaystyle \frac{2\pi }{\lambda }}dsin\theta }\dots e^{j{\displaystyle \frac{2\pi }{\lambda }}d(N-1)sin\theta }]}^T,$$

(3)

where d represents the spacing between adjacent antennas and N is the number of antennas. Also, the array steering vector $\mathbf{a}(\vartheta ,\varphi )$ for a UPA can be expressed as follows:

$$\mathbf{a}(\theta ,\varphi )={\mathbf{a}}_y(\theta ,\varphi )\otimes {\mathbf{a}}_x(\theta ,\varphi ),$$

(4)

where ${\mathbf{a}}_x(\theta ,\varphi )$ is a $N_x\times 1$ column vector with its $n_x$-th element being $e^{j{\displaystyle \frac{2\pi }{\lambda }}d(n_x-1)sin\theta sin\varphi }$ and ${\mathbf{a}}_y(\theta ,\varphi )$ is a $N_y\times 1$ column vector with its $n_y$-th element being $e^{j{\displaystyle \frac{2\pi }{\lambda }}d(n_y-1)sin\theta sin\varphi }$. ⊗ denotes the Kronecker product. $N_x$ and $N_y$, respectively, denote the number of antenna elements aligned along the horizontal and vertical dimensions of the UPA configuration.

Therefore, the total channel matrix received at the MR is then expressed as follows:

$${\mathbf{H}}_{\mathrm{eff}}\left(t\right)=\mathbf{D}\left(t\right)+\sum _{i=1}^{N_s}{\mathbf{G}}_i\left(t\right){\mathbf{\Theta }}_i\left(t\right){\mathbf{M}}_i,$$

(5)

For the convenience of representation, we represent the channel matrix equivalently in a compact form:

$${\mathbf{H}}_{\mathrm{eff}}\left(t\right)=\mathbf{D}\left(t\right)+\mathbf{G}\left(t\right)\mathbf{\Theta }\left(t\right)\mathbf{M},$$

(6)

where $\mathbf{G}\left(t\right)=[{\mathbf{G}}_1\left(t\right),{\mathbf{G}}_2\left(t\right),\dots ,{\mathbf{G}}_{N_s}\left(t\right)]\in {\mathbb{C}}^{N_R\times N}$, $\mathbf{M}={[{\mathbf{M}}_1^T,{\mathbf{M}}_2^T,\dots ,{\mathbf{M}}_{N_s}^T]}^T\in {\mathbb{C}}^{N\times N_T}$, $\mathbf{\Theta }\left(t\right)=[{\mathbf{\Theta }}_1\left(t\right),{\mathbf{\Theta }}_2\left(t\right),\dots ,{\mathbf{\Theta }}_{N_s}\left(t\right)]\in {\mathbb{C}}^{N\times N}$.

We assume that the real-time perfect CSI of all links during the MR movement can be obtained from the centralized controller via existing channel estimation methods [36]. Therefore, the received signal at the MR at time slot t can be expressed as follows:

$$\mathbf{y}\left(t\right)=\left(\sqrt{{\displaystyle \frac{P}{S\left(t\right)}}}\right){\mathbf{H}}_{\mathrm{eff}}\left(t\right)\mathbf{F}\left(t\right)\mathbf{s}\left(t\right)+\mathbf{n}\left(t\right),$$

(7)

where P denotes the total transmit power per time slot, and $\mathbf{n}\left(t\right)\sim \mathcal{CN}(0,{\sigma }^2{\mathbf{I}}_{N_R})$ is an additive Gaussian white noise vector with mean 0 and variance ${\sigma }^2$. To fully exploit the spatial diversity of MIMO during MR movement, we consider $S\left(t\right)=\mathrm{rank}\left({\mathbf{H}}_{\mathrm{eff}}\left(t\right)\right)$. The aim of this paper is to maximize the rate from BS to MR at any moment through the co-optimization of the precoding $\mathbf{F}$ and the phase shift matrix $\mathbf{\Theta }$ of all IRSs. The achievable rate is formulated as follows:

$$R\left(t\right)={log}_{2}det\left({\mathbf{I}}_{N_R}+{\displaystyle \frac{P}{{\sigma }^{2}S\left(t\right)}}{\mathbf{H}}_{\mathrm{eff}}\left(t\right)\mathbf{F}\left(t\right)\mathbf{F}{\left(t\right)}^H{\mathbf{H}}_{\mathrm{eff}}{\left(t\right)}^H\right),$$

(8)

In MIMO systems, the precoding $\mathbf{F}$ can be derived through singular value decomposition (SVD) of the effective channel matrix ${\mathbf{H}}_{\mathrm{eff}}$. More specifically, for a time slot, the SVD of ${\mathbf{H}}_{\mathrm{eff}}$ is expressed as ${\mathbf{H}}_{\mathrm{eff}}=\mathbf{U}\mathbf{\Lambda }{\mathbf{V}}^H$, where $\mathbf{U}\in {\mathbb{C}}^{N_R\times N_R}$ and $\mathbf{V}\in {\mathbb{C}}^{N_T\times N_T}$ are unitary matrices and $\mathbf{\Lambda }\in {\mathbb{R}}^{N_R\times N_T}$ is a diagonal matrix containing ordered singular value, i.e., ${\lambda }_1\ge {\lambda }_2\ge \dots \ge {\lambda }_S$. Based on this decomposition, the optimal precoding at this time slot can be constructed as ${\mathbf{F}}^{\mathrm{opt}}=\mathbf{V}{\widehat{\mathbf{P}}}^{1/2}$, where the power allocation matrix $\widehat{\mathbf{P}}={[\mathrm{diag}(p_1,\dots ,p_S),{\mathbf{0}}_{S\times (N_T-S)}]}^T\in {\mathbb{R}}^{N_T\times S}$ satisfies $p_s\ge 0$, representing the power allocation for each independent data stream, with $s\in \mathcal{S}\triangleq \{1,2,\dots ,S\}$, and the total power constraint ${\sum }_{i=1}^S{p}_i=S$. This method equivalently decomposes the original MIMO channel into S parallel SISO spatial paths, each with a channel gain of ${\lambda }_s$. Therefore, at time slot t, the rate of the s-th eigenchannel can be expressed as $R_s\left(t\right)={log}_2\left(1+{\displaystyle \frac{P{p}_s\left(t\right){\lambda }_s{\left(t\right)}^2}{{\sigma }^{2}S\left(t\right)}}\right)$. The sum-rate maximization problem for the V2I system is thereby transformed into maximizing the total capacity of these equivalent SISO channels, leading to the following optimization formulation:

$$\begin{array}{cc}\hfill \underset{\widehat{\mathbf{P}}\left(t\right),\mathbf{\Theta }\left(t\right)}{max}& R=\sum _{s=1}^{S\left(t\right)}{log}_2\left(1+{\displaystyle \frac{P{p}_s\left(t\right){\lambda }_s{\left(t\right)}^2}{{\sigma }^{2}S\left(t\right)}}\right)\hfill \end{array}$$

(9a)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \mathbf{\Theta }{\left(t\right)}_{ii}=e^{j{\theta }_i},i=1,2,\dots ,N,\hfill \end{array}$$

(9b)

$$\begin{array}{cc}\hfill & \sum _{i=1}^{S\left(t\right)}{p}_i=S\left(t\right),\hfill \end{array}$$

(9c)

Due to the unit modulus constraints of the above problem and the coupling between ${\lambda }_s\left(t\right)$ and $\mathbf{\Theta }\left(t\right)$, the problem is non-convex, making it challenging to obtain the globally optimal phase shift matrix $\mathbf{\Theta }\left(t\right)$ and power allocation matrix $\widehat{\mathbf{P}}\left(t\right)$. Our goal is to address this problem with low computational complexity in order to satisfy the channel time-varying requirements.

3. Problem Solution

For brevity, the time slot index t is omitted in subsequent algorithm derivations. From Equation (9), it can be observed that for any given $\mathbf{\Theta }$, once ${\lambda }_s$ ($\forall s\in \mathcal{S}$) is obtained, the $\widehat{\mathbf{P}}$ can be derived using the WF algorithm. Consequently, our analysis concentrates on the coordinated design of multiple IRSs phase shift matrices. To address the optimization challenge in dynamic channel environments, this paper adopts the sum path gain maximization (SPGM) criterion proposed in [14].

3.1. Phase Shift Matrix Design Under the SPGM Criterion

It is evident that the rate of R monotonically increases with ${\lambda }_s$ ($\forall s\in \mathcal{S}$), implying that R depends on the channel quality of ${\mathbf{H}}_{\mathrm{eff}}$. To maximize the channel quality of ${\mathbf{H}}_{\mathrm{eff}}$, we suggest maximizing the sum gain of spatial paths between BS and MR, i.e., ${\sum }_{s=1}^S{\lambda }_s^2$. Thus, the optimization problem transforms into the following:

$$\begin{array}{cc}\hfill \underset{\mathbf{\Theta }}{max}& \widehat{R}\stackrel{\left(a\right)}{=}\sum _{s=1}^S{\lambda }_s^2\hfill \\ \hfill & \stackrel{\left(b\right)}{=}\mathrm{Tr}\left[{(\mathbf{D}+\mathbf{G}\mathbf{\Theta }\mathbf{M})}^H(\mathbf{D}+\mathbf{G}\mathbf{\Theta }\mathbf{M})\right]\hfill \end{array}$$

(10a)

$$\mathrm{s}.\mathrm{t}.{\mathbf{\Theta }}_{ii}=e^{j{\theta }_i},i=1,2,\dots ,N$$

(10b)

where $\left(b\right)$ holds since ${\lambda }_s^2$ corresponds to the s-th eigenvalue of ${\mathbf{H}}_{\mathrm{eff}}^H{\mathbf{H}}_{\mathrm{eff}}$, leading to ${\sum }_{s=1}^S{\lambda }_s^2=\mathrm{Tr}\left({\mathbf{H}}_{\mathrm{eff}}^H{\mathbf{H}}_{\mathrm{eff}}\right)$.

As noted in [14], this criterion offer a lower bound for the sum rate in Equation (8). Nevertheless, optimizing based on this criterion yields a phase shift matrix that achieves a sum rate near the optimal value. To effectively address this, we generalize the problem into a more comprehensive form:

$$\begin{array}{cc}\hfill \underset{\mathbf{\Theta }}{max}& \widehat{R}=\mathrm{Tr}\left[{(\mathbf{D}+\mathbf{G}\mathbf{\Theta }\mathbf{M})}^H(\mathbf{D}+\mathbf{G}\mathbf{\Theta }\mathbf{M})\right]\hfill \end{array}$$

(11a)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& {\mathbf{\Theta }}_{ij}=e^{j{\theta }_{ij}},i,j=1,2,\dots ,N,\hfill \end{array}$$

(11b)

Here, $\mathbf{\Theta }\in {\mathbb{C}}^{N\times N}$ is no longer a diagonal matrix, but a full matrix, thus applying to a wider range of scenarios. We derive for $\widehat{R}$ with respect to ${\theta }_{ij}$:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \widehat{R}}{\partial {\theta }_{ij}}}& \stackrel{\left(a\right)}{=}\mathrm{tr}\left[{\mathbf{M}}^H{\displaystyle \frac{\partial {\mathbf{\Theta }}^H}{\partial {\theta }_{ij}}}{\mathbf{G}}^H(\mathbf{D}+\mathbf{G}\mathbf{\Theta }\mathbf{M})\right]+\mathrm{tr}\left[{(\mathbf{D}+\mathbf{G}\mathbf{\Theta }\mathbf{M})}^H\mathbf{G}{\displaystyle \frac{\partial \mathbf{\Theta }}{\partial {\theta }_{ij}}}\mathbf{M}\right]\hfill \\ \hfill & \stackrel{\left(b\right)}{=}2\Re \mathrm{e}\left\{j{e}^{j{\theta }_{ij}}{(\mathbf{D}+\mathbf{G}\mathbf{\Theta }\mathbf{M})}^H\mathbf{G}{\mathbf{e}}_j{\mathbf{e}}_i^T\mathbf{M}\right\}\hfill \\ \hfill & \stackrel{\left(c\right)}{=}-2\Im \mathrm{m}\left\{e^{j{\theta }_{ij}}{\left[\mathbf{M}{(\mathbf{D}+\mathbf{G}\mathbf{\Theta }\mathbf{M})}^H{\mathbf{G}}^T\right]}_{ij}\right\},\hfill \end{array}$$

(12)

where ${\mathbf{e}}_i$ and ${\mathbf{e}}_j$ are the standard basis vector of dimension N, with the i-th element being 1 and others 0. $\left(b\right)$ utilizes the cyclic invariance of the trace and the property of conjugate transpose. $\left(c\right)$ follows from $\Re \mathrm{e}\left\{jz\right\}=-\Im \mathrm{m}\left\{z\right\}$ for any complex number z. The phase shift matrix $\mathbf{\Theta }$ can be further decomposed into $\mathbf{\Theta }=\tilde{\mathbf{\Theta }}{\left[n\right]}_{ij}+e^{j{\theta }_{ij}}{\mathbf{E}}_{ij}$, where ${\mathbf{E}}_{ij}={\mathbf{e}}_j{\mathbf{e}}_i^T$, and $\tilde{\mathbf{\Theta }}$ retains all elements of $\mathbf{\Theta }$ except the $(i,j)$-th entry, which is set to zero. Thus, the first-order derivative can be expressed as

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \widehat{R}}{\partial {\theta }_{ij}}}& \stackrel{\left(a\right)}{=}-2\Im \mathrm{m}\left\{e^{j{\theta }_{ij}}{\left[\mathbf{M}{(\mathbf{D}+\mathbf{G}{\tilde{\mathbf{\Theta }}}_{ij}\mathbf{M})}^H\mathbf{G}+e^{-j{\theta }_{ij}}\mathbf{M}{\left(\mathbf{G}{\mathbf{E}}_{ij}\mathbf{M}\right)}^H\mathbf{G}\right]}_{ij}\right\}\hfill \\ \hfill & \stackrel{\left(b\right)}{=}-2\Im \mathrm{m}\left\{e^{j{\theta }_{ij}}{\left[{\mathbf{G}}^T{(\mathbf{D}+\mathbf{G}{\tilde{\mathbf{\Theta }}}_{ij}\mathbf{M})}^{\ast }{\mathbf{M}}^T\right]}_{ij}\right\}\hfill \\ \hfill & \stackrel{\left(c\right)}{=}-2\Im \mathrm{m}\left\{e^{j{\theta }_{ij}}{z}_{ij}\right\},\hfill \end{array}$$

(13)

where $\left(b\right)$ is because ${\left[\mathbf{M}{\left(\mathbf{G}{\mathbf{E}}_{ij}\mathbf{M}\right)}^H\mathbf{G}\right]}_{ij}={\mathbf{e}}_j^T\mathbf{M}{\left(\mathbf{G}{\mathbf{e}}_i{\mathbf{e}}_j^T\mathbf{M}\right)}^H\mathbf{G}{\mathbf{e}}_i={\left[\mathbf{M}{\mathbf{M}}^H\right]}_{jj}{\left[{\mathbf{G}}^H\mathbf{G}\right]}_{ii}\triangleq r_{ij}$, which is real-valued, and the property $\Im \mathrm{m}\left\{c\right\}=\Im \mathrm{m}\{c+r_{ij}\}$ for any complex number c, $z_{ij}={\left[{\mathbf{G}}^T{(\mathbf{D}+\mathbf{G}{\tilde{\mathbf{\Theta }}}_{ij}\mathbf{M})}^{\ast }{\mathbf{M}}^T\right]}_{ij}$. From Equation (13), $z_{ij}$ depends on all angles in $\mathbf{\Theta }$ except ${\theta }_{ij}$. Furthermore, when all angles except ${\theta }_{ij}$ are fixed, $\partial \widehat{R}/\partial {\theta }_{ij}$ is a sinusoidal function of ${\theta }_{ij}$ with a period of $2\pi$. Consequently, $\widehat{R}$ at time slot t also exhibits sinusoidal behavior with respect to ${\theta }_{ij}$, sharing the same period of $2\pi$. Based on the properties of sinusoidal functions, $\widehat{R}$ has one global maximum and one global minimum within each period. Setting $\partial \widehat{R}/\partial {\theta }_{ij}=0$ yields the critical points:

$$\begin{array}{cc}\hfill {\theta }_{ij}& =\angle z_{ij}^{\ast }+2\pi k={\left[{\mathbf{G}}^T{(\mathbf{D}+\mathbf{G}{\tilde{\mathbf{\Theta }}}_{ij}\mathbf{M})}^{\ast }{\mathbf{M}}^T\right]}_{ij}+2\pi k,\hfill \end{array}$$

(14a)

$$\begin{array}{cc}\hfill {\theta }_{ij}& =\angle z_{ij}^{\ast }+\pi +2\pi k={\left[{\mathbf{G}}^T{(\mathbf{D}+\mathbf{G}{\tilde{\mathbf{\Theta }}}_{ij}\mathbf{M})}^{\ast }{\mathbf{M}}^T\right]}_{ij}+\pi +2\pi k,\hfill \end{array}$$

(14b)

where k is an integer. To determine the maximum point, we compute the second-order derivative of $\widehat{R}$ with respect to ${\theta }_{ij}$:

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^2\widehat{R}}{\partial {\theta }_{ij}^2}}& \stackrel{\left(a\right)}{=}2\Im \mathrm{m}\left\{-e^{j{\theta }_{ij}}{\mathbf{e}}_j^T\mathbf{M}{(\mathbf{D}+\mathbf{G}{\mathbf{\Theta }}_{ij}\mathbf{M})}^H\mathbf{G}{\mathbf{e}}_i+j{e}^{j{\theta }_{ij}}{\mathbf{e}}_j^T\mathbf{M}{\mathbf{M}}^H(-j{e}^{-j{\theta }_{ij}}){\mathbf{E}}_{ij}^H\mathbf{G}{\mathbf{e}}_i\right\}\hfill \\ \hfill & \stackrel{\left(b\right)}{=}2\Im \mathrm{m}\left\{-e^{j{\theta }_{ij}}{\left[\mathbf{M}{(\mathbf{D}+\mathbf{G}{\mathbf{\Theta }}_{ij}\mathbf{M})}^H\mathbf{G}\right]}_{ji}+{\left[\mathbf{M}{\mathbf{M}}^H\right]}_{jj}{\left[{\mathbf{G}}^H\mathbf{G}\right]}_{ii}\right\}\hfill \\ \hfill & \stackrel{\left(c\right)}{=}-2\Re \mathrm{e}\left\{e^{j{\theta }_{ij}}{\left[\mathbf{M}{(\mathbf{D}+\mathbf{G}{\mathbf{\Theta }}_{ij}\mathbf{M})}^H\mathbf{G}\right]}_{ji}\right\}-2\Re \mathrm{e}\left\{e^{j{\theta }_{ij}}{\left[\mathbf{M}{\left(\mathbf{G}{e}^{j{\theta }_{ij}}{\mathbf{E}}_{ij}\mathbf{M}\right)}^H\mathbf{G}\right]}_{ji}\right\}\hfill \\ \hfill & +2\Re \mathrm{e}\left\{{\left[\mathbf{M}{\mathbf{M}}^H\right]}_{jj}{\left[{\mathbf{G}}^H\mathbf{G}\right]}_{ii}\right\}\hfill \\ \hfill & \stackrel{\left(d\right)}{=}-2\Re \mathrm{e}\left\{e^{j{\theta }_{ij}}{z}_{ij}\right\},\hfill \end{array}$$

(15)

where (d) holds because the second and third terms in (c) offset one another (as proven in the first-order derivative derivation). Since the maximum point lies in the concave interval, the condition ${\partial }^2\widehat{R}/\partial {\theta }_{ij}^2<0$ must be satisfied. Therefore, the optimal ${\theta }_{ij}$ at time slot t that maximizes $\widehat{R}$ requires $\Re \mathrm{e}\left\{e^{j{\theta }_{ij}}{z}_{ij}\right\}>0$. Substituting Equation (14a) into the above yields the following:

$$\Re \mathrm{e}\left\{e^{j{\theta }_{ij}}{z}_{ij}\right\}=\Re \mathrm{e}\left\{e^{j{\theta }_{ij}}\mid z_{ij}\mid e^{-j{\theta }_{ij}}\right\}=\mid z_{ij}\mid >0,$$

(16)

That is, Equation (14a) is the point of local maximum. Similarly, Equation (14b) is the point of local minimum. Since Equation (11) is an extended version of Equation (10), and the matrix $\mathbf{\Theta }$ in (10) is diagonal, we iteratively update the i-th diagonal element ${\theta }_i$ ($i=1,2,\dots ,N$) using Equation (14a). More specifically, at time t, during each iteration, we fix other diagonal elements ${\left\{{\theta }_j\right\}}_{j\ne i}$ and update ${\theta }_i$ via a closed-form analytical solution. This process iteratively optimizes the mutually coupled variables ${\theta }_i$ in the phase shift matrix $\mathbf{\Theta }$, where the optimal ${\theta }_i$ in each iteration can be explicitly derived. Consequently, the optimal phase configuration for each IRS element is formulated as follows:

$${\theta }_i^{\left(k\right)}=\angle \left[d_{ii}+\sum _{\widehat{i}=1}^{i-1}exp\left(j{\theta }_i^{\left(k\right)}\right)g_{i\widehat{i}}{m}_{\widehat{i}i}+\sum _{\widehat{i}=i+1}^{N}exp\left(j{\theta }_i^{(k-1)}\right)g_{i\widehat{i}}{m}_{\widehat{i}i}\right],$$

(17)

where $d_{ii}\triangleq {\left[{\mathbf{G}}^H\mathbf{D}{\mathbf{M}}^H\right]}_{ii}$, $g{\left[n\right]}_{i\widehat{i}}\triangleq [{\mathbf{G}}^H{\mathbf{G}}_{i{i}^{\prime }}$, $m_{\widehat{i}i}\triangleq {\left[\mathbf{M}{\mathbf{M}}^H\right]}_{\widehat{i}i}$, $\forall i,\widehat{i}\in \{1,2,\dots ,N\}$, and k represents the iteration index. This algorithm leverages a critical property: for a given t, when fixing ${\left\{{\theta }_j\right\}}_{j\ne i}$, the matrix $\mathbf{\Theta }$ exhibits sinusoidal behavior with respect to ${\theta }_i$. And the principle is similar to the famous block coordinate descent (BCD) method: fix the other diagonal elements ${\left\{{\theta }_j\right\}}_{j\ne i}$ and maximize $\widehat{R}$ by alternately optimizing ${\theta }_i$. Therefore, we name this algorithm ’dimension-wise sine maximization’ (DSM).

3.2. Precoding Design with the Obtained Power Allocation Matrix

At time t, the optimal $\widehat{\mathbf{P}}$ can be obtained by resolving Equation (9) through the WF algorithm [37], given the designed phase shift matrix. More specifically, for $\forall s\in \mathcal{S}$, the optimal $p_s$ is determined as follows:

$$p_s=max\left(\zeta -{\displaystyle \frac{{\sigma }^{2}S}{P{\lambda }_s^2}},0\right),$$

(18)

where $\zeta$ is a constant to ensure ${\sum }_{i=1}^S{p}_i=S$, the value of which can be calculated numerically. Once $\widehat{\mathbf{P}}$ is obtained, the optimal precoding matrix ${\mathbf{F}}^{\mathrm{opt}}=\mathbf{V}{\widehat{\mathbf{P}}}^{1/2}$ can be directly constructed, where $\mathbf{V}$ is the unitary matrix derived from the SVD of ${\mathbf{H}}_{\mathrm{eff}}$. The complete algorithm for solving Equation (9) is encapsulated in Algorithm 1.

Algorithm 1: Proposed DSM Algorithm to Solve (9)

As shown in Algorithm 1, at any time slot t, the proposed method operates through two sequential steps: first, by computing the optimal phase shift matrix $\mathbf{\Theta }$ via the DSM algorithm (Lines 3–10), and second, by determining the $\widehat{\mathbf{P}}$ and the $\mathbf{F}$ using the WF algorithm (Line 12). The DSM algorithm requires N iterations of Equation (8) (Lines 5–7), with each iteration incurring $\mathcal{O}\left(N\right(N-1\left)\right)$ operations, while the WF algorithm contributes a lower complexity of $\mathcal{O}\left(N\right)$. Consequently, the overall complexity of Algorithm 1 scales as $\mathcal{O}\left(N^2\right)$, primarily driven by the phase shift optimization. For comparison,

Table 2. The computational complexity comparison.

Literature	Method	Complexity
[7]	Semidefinite relaxation (SDR)	$\mathcal{O}\left(N^6\right)$
[14]	Alternating direction method of multipliers (ADMM)	$\mathcal{O}\left(N^3\right)$
This paper	DSM	$\mathcal{O}\left(N^2\right)$

summarizes existing optimization methods based on the SPGM criterion in [7,14] and their corresponding complexities. From Table 2, it can be observed that the low complexity of the proposed algorithm enables it to better adapt to channel variations in real time, contributing to stable and high-quality communication in high-speed vehicular mobility scenarios.

4. Results

This section presents numerical simulations to evaluate the effectiveness of the DSM algorithm in multi-IRS-assisted V2I systems. The initial layout is shown in Figure 2. Unless otherwise stated, the MR moves at a speed of $v_R=10\mathrm{m}/\mathrm{s}$ with an azimuth angle ${\gamma }_R={30}^{\circ }$, which leads to a a maximum Doppler shift $f_{max}=v_R{f}_c/c=80\mathrm{Hz}$. Consequently, the time slot length is set to $T_b=1/10f_{max}\approx 1\mathrm{ms}$. To ensure clarity in our analysis, the communication process is evaluated over a 5-second period, meaning $T=⌈{\displaystyle \frac{5}{T_b}}⌉$. In addition, the other primary simulation parameters are shown in

Table 3. Simulation parameters.

Parameter	Value
Carrier frequency: $f_c$	2.4 GHz
Number of transmit antennas: $N_T$	16
Number of receive antennas: $N_R$	12
Number of IRSs: $N_S$	2
Number of IRS reflection elements: $N_{IRS}$	64
Antenna and IRS element spacing: $\delta$	$\lambda /2$
Transmit power at the BS: P	20 dBm
Noise power at the MR: ${\sigma }^2$	0 dBm
BS location	(0 m, 0 m, 2 m)
IRS1 location	(30 m, 40 m, 8 m)
IRS2 location	(40 m, 40 m, 8 m)
Reference loss at 1 m: $\rho$	30 dB
Path-loss exponents: $a_{\mathrm{BS}-\mathrm{MR}}$, $a_{\mathrm{IRS}-\mathrm{MR}}$, and $a_{\mathrm{BS}-\mathrm{IRS}}$	3, 2.8, and 2.2
Rician factor: $\kappa$	4 dB

. The average sum rate, defined as $\overline{R}={\displaystyle \frac{1}{T}}{\sum }_{t=1}^{T}R\left(t\right)$, serves as the primary performance metric.

The convergence of Algorithm 1 at the time slot $t=0$ is first analyzed, as shown in Figure 3a, with the termination criterion $ϵ={10}^{-2}$. The DSM algorithm exhibits better convergence characteristics than the ADMM algorithm under all transmit power conditions. More specifically, the sum rate of the DSM algorithm stabilizes more rapidly and achieves higher values than ADMM. For example, at a transmission power of 20 dBm, the DSM algorithm achieves a stabilized rate approximately 14.9% higher than ADMM with significantly fewer iterations. The inferior performance of ADMM in Figure 3a may stem from the coupling effects in multi-IRS-assisted channels; while ADMM optimizes the phase shift matrices by treating multiple IRSs as a single entity, the DSM algorithm independently optimizes each phase shift element, enabling better adaptability to multi-IRS scenarios. Furthermore, Figure 3b compares the relationship between the average number of iterations and the number of per IRS reflecting elements $N_{IRS}$ under different convergence criteria for both algorithms. It is observed that as $N_{IRS}$ increases, the average iteration count for both algorithms rises. However, under the same convergence criterion and $N_{IRS}$, the DSM algorithm requires a significantly lower average number of iterations than the ADMM algorithm. Therefore, the DSM algorithm proposed in this paper not only has lower complexity, but also has far fewer iterations than the ADMM algorithm, which is better adapted to the rapid change of the channel in the V2I system.

Figure 4 illustrates the correlation between the $\overline{R}$ and the transmit power. To evaluate the performance of the DSM algorithm based on the SPGM criterion, we compare it with four baseline schemes: (1) randomly generated phase shift scheme; (2) exhaustive search of 50,000 random phase shift schemes to solve Equation (8); (3) DSM with 1/2-bit IRS; and (4) without IRS. For all baselines, the determination of the precoding design is consistent with Section 3.2. As shown in Figure 4, the DSM algorithm achieves a performance comparable to exhaustive search but with substantially reduced computational complexity, further validating the effectiveness of the SPGM criterion and the DSM algorithm. Furthermore, the multi-IRS-aided V2I system demonstrates a remarkable capacity advantage over the system without IRS, attributed to the channel quality improvement achieved by IRS through beamforming gain and inherent aperture gain. Practical implementations with discrete phase shifts (1/2-bit) achieve near-optimal performance despite quantization losses, as evidenced by the experimental results. For instance, when $N_{IRS}=64$, the 2-bit configuration incurs a mere 5.1% performance loss compared to continuous phase shifts.

Figure 5 investigates the impact of the number of IRS reflecting elements $N_{IRS}$ on $\overline{R}$. As $N_{IRS}$ increases, the sum rate of the system assisted by multiple IRSs increases, while the system without IRS maintains a lower value unaffected by $N_{IRS}$. This enhancement comes from the increased signal reflection capability provided by additional elements, which enhances the passive beamforming effect. Furthermore, the disparity in performance between the DSM-based continuous phase shift scheme and the 1-bit/2-bit discrete phase shift schemes widens as $N_{IRS}$ grows. This phenomenon arises because more reflecting elements amplify the accumulation of quantization errors in discrete phase shift configurations, thereby degrading their performance.

Figure 6 illustrates the impact of different Rician factors $\kappa$ on $\overline{R}$. All evaluated schemes exhibit a declining trend as $\kappa$ increases. This occurs because a higher $\kappa$ value amplifies the stronger dominance of the LoS component over the NLos component [38], which increases channel correlation, reduces the number of transmittable data streams, and consequently leads to a sum-rate reduction. In particular, when $\kappa$ approaches infinity, regardless of the design of the phase shift matrix, the effective MIMO channel has a rank of one, allowing support for only a single data stream. Therefore, in practical deployments, placing the IRS in richly scattered environments can efficiently boost the rank of the MIMO channel, and thus enhance the spatial multiplexing gain.

Figure 7 explores the relationship between the $\overline{R}$ and the MR mobile azimuth ${\gamma }_R$. It is found that when ${\gamma }_R$ varies in the range of −90° to 90°, the sum rate of the DSM-based scheme decreases and then increases, while the sum rate of the system without IRS nearly always remains at a relatively low level. More specifically, when the MR moves from the negative y-axis toward the x-axis, the increasing angle causes the MR to gradually move away from both the BS and the IRS, intensifying path loss and consequently reducing capacity. However, as ${\gamma }_R$ further increases, the MR is relatively close to the IRS, and the IRS-assisted indirect link plays a dominant role, which makes the capacity increase accordingly. Compared with ${\gamma }_R={30}^{\circ }$, the V2I sum rate decreases and increases by 6.51% and 16.11% at ${\gamma }_R=0^{\circ }$ and ${\gamma }_R={60}^{\circ }$, respectively. This suggests that the direction of vehicle motion is crucial to guarantee high-quality communication in complex urban environments [39].

Figure 8 illustrates the real-time sum-rate variation using the DSM algorithm during MR motion from 0 to 5 s. Although movement-induced path loss leads to a declining trend in the rate, the DSM algorithm effectively meets communication demands by optimizing the instantaneous rate at each time slot. Due to the difficulty in obtaining perfect real-time CSI during MR mobility, Figure 8 also illustrates the real-time sum rate under imperfect CSI conditions. In our model, the true channel matrix is derived from the instantaneous position of the vehicle. We assume that the estimated channel matrix consists of the sum of the true and error matrices, and the error matrix elements are independently and identically distributed, following the $\mathcal{CN}(0,{\sigma }^2)$ distribution. It can be seen that at $t=2$ s, even if ${\sigma }^2$ is relatively large (e.g., ${\sigma }^2=0.9$), the sum rate decreases by only 3.63 bits/Hz compared to the perfect CSI scenario. Therefore, the proposed optimization algorithm remains effective even under imperfect CSI. In addition, thanks to the low complexity of the algorithm, the V2I system can achieve adaptive control according to the real-time position of the vehicle and the environmental variations through this dynamic optimization approach.

We evaluate the impact of the number of IRS on the $\overline{R}$. In the scenario shown in Figure 9, each IRS is equipped with 64 reflecting elements. The first IRS is fixed at coordinates (30, 40, 8), while the coordinates of subsequent IRSs are incrementally shifted along the x-axis by a distance $d_{IRS}$ relative to the previous one. As illustrated in Figure 9a, increasing the number of IRSs enables the system to better exploit spatial multiplexing gains and achieve superior performance improvements. For Figure 9b, when the total number of IRS reflecting elements is fixed at 64, a single IRS almost achieves a similar performance gain to multiple IRSs, which is consistent with the findings in [40]. However, as the transmit power increases, the capacity enhancement from multi-IRS configurations becomes significantly more pronounced than that from a single IRS. For example, at a transmit power of 20 dBm, the multi-IRS configuration yields an 8.7% performance gain compared to a single IRS. This suggests that distributed IRS deployment enhances signal coverage through spatial diversity, where multiple IRSs reflect signals from distinct locations, thereby improving channel multiplexing efficiency.

Considering the critical role of IRS deployment in real environments on system communication performance, we also investigated the effect of the spacing between two IRSs $d_{IRS}$ on the $\overline{R}$ of the V2I system. Initially, both IRSs are positioned at coordinates (50, 40, 8). As $d_{IRS}$ increases, the spacing between the two IRSs increases, with one IRS moving closer to the BS and the other approaching the MR. From Figure 10, it can be noticed that the increase in $d_{IRS}$ causes a rise in the sum rate. More specifically, when $d_{IRS}$ = 100 m, the sum rate is 13.5% higher compared to when $d_{IRS}$ = 0 m, which is expected. When IRSs are deployed near the BS or user area, this effectively mitigates the dual path loss and thus improves the gain. This confirms that optimal IRS placement should prioritize proximity to either the transmitter or receiver.

5. Conclusions

In this paper, we investigate the sum rate maximization problem for multi-IRS-assisted V2I time-varying communication systems. To address this challenge, we propose a low-complexity DSM algorithm based on the SPGM criterion for optimizing the all-IRS phase shift matrix. Utilizing the optimized IRS phase shift matrices, the WF algorithm is used to determine the BS precoding. The DSM algorithm follows the same principle as the BCD method: the target variables (the phase shift matrix) are first decoupled, and then the phase of each reflecting element is optimized one by one. Simulation results demonstrate that, compared with the traditional ADMM, the sum rate of the DSM algorithm is increased by 14.9%. While ensuring communication quality, the DSM algorithm has a faster convergence speed and is thus suitable for real-time optimization in high-speed mobile scenarios. Additionally, the multi-IRS auxiliary mode better leverages spatial diversity characteristics, yielding an 8.7% gain over the single-IRS configuration with superior performance. Moreover, simulation results also verify that the vehicle movement direction and IRS deployment location significantly impact system performance in practical V2I systems. Considering the difficulty of obtaining perfect CSI, future research should focus on robust optimization algorithms under imperfect CSI.