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Hybrid precoding schemes for mmWave massive MIMO V2V system with finite-resolution PSs
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Xiaolin Liang1, 2, Zhanyi Rong1, Wangbin Cao3
The Journal of China Universities of Post and Telecommunications | 2025, 32(2) : 70 - 81
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The Journal of China Universities of Post and Telecommunications | 2025, 32(2): 70-81
• 6G Technological Innovation and Future Industrial Development •
Hybrid precoding schemes for mmWave massive MIMO V2V system with finite-resolution PSs
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Xiaolin Liang1, 2, Zhanyi Rong1, Wangbin Cao3
Affiliations
  • 1College of Electronic and Information Engineering, Hebei University, Baoding 071002, China
  • 2Hebei Key Laboratory of Power Internet of Things Technology, North China Electric Power University, Baoding 071003, China
  • 3School of Electrical and Electronic Engineering, North China Electric Power University, Baoding 071003, China
doi: 10.19682/j.cnki.1005-8885.2025.0017
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Considering the significant expenses and power requirements associated with digital precoding and the low spectral efficiency (SE) of analog precoding, a hybrid precoding algorithm with efficient finite-resolution phase shifters (PSs), named FRPS algorithm, was proposed for millimeter wave (mmWave) massive multi-input multi-output (MIMO) vehicle-to-vehicle (V2V) system. Digital and analog precoding matrix variables and non-convex constraints of the radio frequency (RF) precoder are decomposed into two independent optimization problems. Discrete iterative optimization is employed to solve for the analog and digital precoder alternately. In addition, the effect of finite-resolution PSs is considered in the precoding algorithm optimization. Simulation results show that the FRPS algorithm has a fast convergence speed, and it can approach the SE of fully digital precoding with only 3 bit resolution PS. The performance difference between the FRPS algorithm and the existing hybrid precoding algorithm with infinite-resolution PSs is almost negligible. Moreover, the FRPS algorithm performs superior to that of the infinite-resolution PS hybrid precoding algorithm and fully digital precoding algorithm.

hybrid precoding  /  millimeter wave (mmWave) communication  /  vehicle-to-vehicle (V2V) communication  /  finite-resolution phase shifters  /  massive multi-input multi-output (MIMO)
Xiaolin Liang, Zhanyi Rong, Wangbin Cao. Hybrid precoding schemes for mmWave massive MIMO V2V system with finite-resolution PSs[J]. The Journal of China Universities of Post and Telecommunications, 2025 , 32 (2) : 70 -81 . DOI: 10.19682/j.cnki.1005-8885.2025.0017
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1　Introduction
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V2V communication brings profound changes that greatly enhance user experience and improve road safety in the 6th-generation mobile communication (6G) era[1]. As one of the fundamental technologies for intelligent transport systems and autonomous driving[2], V2V communication necessitates elevated standards for quality of service (QoS) and reliability[3]. Emerging smart transportation and autonomous driving systems require significantly higher bandwidth and data rates than those currently supported by sub-6GHz vehicle-to-everything (V2X) technologies. This triggers interest in developing multiple antenna technologies and mmWave precoding schemes that enable V2V communication to transmit data in the desired direction[4].
The mmWave communication has the characteristics of large bandwidth, strong directionality, and exact signal delivery. It is vital for solving the problem of insufficient bandwidth in current mobile communication systems[5-6]. Despite challenges such as path loss and weather-related interference[7], mmWave signals enable the integration of a large number of antenna elements into a small form factor[8]. Merging mmWave transmission with massive MIMO is a widely discussed issue[9]. Large-scale antenna arrays on vehicle transceivers help to mitigate the path and penetration losses in the mmWave band [10].
The mmWave MIMO precoding is used in V2V systems to provide significant beamforming gains. A digital precoder is usually used for traditional MIMO precoding. However, if a digital precoder is used, each antenna requires a separate RF chain. The increase of the RF chain will bring exorbitant costs and energy expenditure[11]. Analog precoders implemented through PS can solely manipulate the signal phase, resulting in significant performance degradation. In mmWave massive MIMO systems, hybrid precoding has emerged as a promising approach, capable of achieving performance comparable to fully digital precoders, while significantly reducing the number of required RF chains[12-15].
In hybrid precoding, the transmitting and receiving analog and digital precoders are coupled. This coupling renders the optimized objective function non-convex and arduous to solve[16]. To address the non-convex challenge, hybrid precoding can be reformulated as a matrix factorization issue. The Euclidean distance between hybrid and fully digital precoding is then minimized. Using the sparsity of mmWave channels and the orthogonal matching pursuit (OMP) method, an approximate optimal unconstrained precoding algorithm was proposed in Ref. [17]. A manifold optimization based alternating minimization (MO-AltMin) algorithm was proposed in Ref. [18]. Initially, the method breaks down the primary issue into combinations involving hybrid precoding and subproblems, subsequently concentrates on resolving the constant modulus limitations within these subproblems. In Ref. [19], practical yet efficient statistically-aided codebook-based hybrid precoding schemes were presented, which require lower channel feedback overhead.
The above precoder designs all assume infinite-resolution PSs. However, the infinite-resolution PS design is challenging and finite-resolution PSs are often used in practice. When the hybrid precoder uses 1 bit PS, the precoding problem takes on a lattice structure. According to lattice theory, the mean square error performance gap between the fully and hybrid precoder/combiner under the 1 bit PS constraint was deduced in Ref. [20]. An analog precoder and FRPS combination scheme for multiuser MIMO systems was proposed in Ref. [21], which reformulated the analog precoder and combiner design problem as a phase categorization problem. A general neural network architecture was also proposed. A FRPS algorithm with low transmitting power was presented in Ref. [22] , which achieves satisfactory performance in partially connected structures. An iterative phase matching algorithm was proposed in Ref. [23]. Findings demonstrate that the iterative phase matching algorithm can enhance SE in scenarios with finite-resolution.
Investigations on hybrid precoding are mostly for traditional cellular networks. Although mmWave has been extensively utilized in static scenarios, it is still a major challenge to apply mmWave hybrid precoding in high-speed mobile scenarios, because the transceiver moves at high-speed and the channel is different from the static channel[7]. Unmanned aerial vehicle hybrid beamforming for massive MIMO was investigated in Ref. [24], and an approximate closed-form expression for the rate was derived. A reinforcement learning algorithm utilizing the deep deterministic policy gradient method was proposed in Ref. [25]. The algorithm is used to optimize base station (BS) and intelligent reflecting surface (IRS) beamforming to improve vehicle-to-infrastructure (V2I) communication performance. The comprehensive analysis of V2V networks in mmWave was conducted in Ref. [26], and the results demonstrate that proper precoding design can ensure higher communication performance. A MIMO hybrid beamforming combined with a time difference of arrival (TDOA)/frequency difference of arrival (FDOA) mmWave vehicle location method was proposed in Ref. [27]. Most existing work on V2V hybrid precoding assumes the utilization of infinite-resolution PSs, which is unrealistic. Therefore, the hybrid precoding schemes with finite-resolution PSs for mmWave massive MIMO V2V system need further investigation.
To make up for these deficiencies, a hybrid precoding scheme for mmWave massive MIMO V2V system with finite-resolution PSs, FRPS algorithm, is proposed. The digital and analog precoders are collaboratively designed to enhance SE and energy efficiency (EE). The concept of iterative optimization in discrete form is deployed to tackle the analog and digital precoders alternately. A two-stage iterative hybrid precoding algorithm based on finite-resolution PSs is proposed. The main contributions of this paper are summarized as follows.
1) FRPS algorithm is proposed for mmWave massive MIMO V2V system, and it performs well for high-speed mobility scenarios. In order to derive the digital and analog precoding matrix variables and meet the non-convex constraints of the RF precoder, the issue is decomposed into two independent optimization problems, and discrete iterative optimization is employed to alternate optimization between the analog and digital precoders.
2) Aiming to optimize the EE of the algorithm, and avoid the complicated design of infinite-resolution PS, the low-precision quantization model is considered in designing the hybrid precoding algorithm. Then the objective function of hybrid precoding for mmWave massive MIMO V2V system can be simplified to a closed-form solution.
3) The FRPS algorithm is compared to the existing hybrid, analog, and fully digital precoding algorithms. The findings indicate that the algorithm exhibits swift convergence and can improve the SE and EE effectively. Moreover, the FRPS algorithm does not need a large PS resolution, and the system performance can achieve the best when 3 bit resolution PS.
The rest of this paper is organized as follows. Sect. 2 describes the V2V system and channel model. In Sect. 3, the mmWave massive MIMO FRPS algorithm in V2V system is described. Sect. 4 displays the experimental results and comparative analysis of the results. Finally, the conclusions are drawn in Sect. 5.
Notation: A is a matrix, a is a vector. Am,n is the entry on the mth row and nth column of a matrix. andto represent the set of real numbers and complex numbers, respectively. (·)*, (·)T, and (·)H stands for the conjugate, transpose, and conjugate transpose, respectively. det(·) and ‖·‖F denote the determinant and Frobenius norm. tr(·) and vec(·) indicate the trace and vectorization； (·)†is the Moore-Penrose pseudo inverse.[·] denotes the expectation operator. Real part of a complex variable is noted by Re [·]. ◦ and ⊗ denote the Hadamard and Kronecker products between two matrices.
2　System and channel model
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In this section, the mmWave massive MIMO V2V system model and the mmWave channel model with mobility are introduced.
2.1　System model
A mmWave massive MIMO V2V system using a hybrid precoder and combiner with finite-resolution PSs is considered, as shown in Fig. 1. Equipped with MT antenna elements and RF chains, the transmitting vehicle user equipment (t-VUE) is capable of simultaneously dispatching Ns data streams via the mmWave channel to the receiving vehicle user equipment (r-VUE). The r-VUE is equipped with MR antenna elements and RF chains. The conditions and are established to facilitate multi-stream transmission, while minimizing the RF chain count.
At the t-VUE, the digital precoder PD is firstly applied to process the Ns data streams represented by vector s with , represents the identity matrix. To enhance beamforming performance, an analog precoder PRF made up of adjustable PSs is used before the RF signals are emitted by its MT antennas. Therefore, the transmitted signal can be expressed as x = PRFPDs. So, the hybrid precoders , where and PRF∈. Moreover, the normalized transmit power constraint is expressed as. Similarly, the r-VUE employs a hybrid combiner Q = QRFQD∈ to handle the incoming signal, where QD ∈ and are the digital combiner and the analog combiner, respectively. The analog combiner QRF is likewise realized through an set of adjustable PSs. Thus, the received signal is presented as
where ρ denotes the average received power, H ∈ denotes the mmWave massive MIMO channel matrix, and represents additive Gaussian noise vector whose entries are independent and identically distributed (i. i. d.) as . It is presumed that comprehensive channel state information (CSI) is accessible to both the transmitter (Tx) and receiver (Rx)[18,27].
2.2　Channel model
A mmWave massive MIMO V2V network scenario is considered. Specifically, to account for a practical environment, each VUE has the functionality to establish a directional beam, facilitating the conveyance of signals to a counterpart VUE. One acts as a Tx and the other acts as a Rx.
The Saleh-Valenzuela (SV) channel model[28] is considered, which is a conventional channel model in the case of mmWave massive MIMO systems. In addition, the Doppler shifts are further considered because of the mobility of V2V systems. The mmWave propagation channel is characterized by a geometry-based channel model. It is delineated by NC, each composed of NR rays. The MR × MT channel matrix H can be represented as
where t denotes time slot and τ denotes time delay, αil denotes the gain of the lth ray in the ith propagation cluster. αil is assumed to be i. i. d. and follow the distribution , where is the average power of the ith cluster with ,Γisa normalization factor for ensuring that = MTMR. In addition, and represent the receiving and transmitting array response vectors, where , , , stand for azimuthal angle of arrival (AoA), azimuthal angle of departure (AoD), elevational AoA, and elevational AoD of the lth path in the ith cluster, respectively. The azimuth and elevation AoAs and AoDs follow the Laplacian distribution. Uniform planar arrays (UPA) with × antenna elements of mmWave systems are considered. The vector that represents the array response for the lth ray within the ith cluster can be written as
where d denotes the antenna spacing and λ represents the wavelength of the signal. and 0≤q ＜ are element indices. UPAs are used at both t-VUE and r-VUE. A two-dimensional (2D) coordinate system is considered where the position of t-VUE is at the origin (0,0). It is assumed that both the t-VUE and the r-VUE are moving along the horizontal axis with speed vTx and vRx, respectively.
The propagation delay for the (i,l)th path is given by τil =ril/c, with c being the velocity of light and ril is the propagation length with the (i,l)th path. It can be obtained as
where ri is the distance from the Tx of the scatterer in the ith cluster and D is the distance between the t-VUE and r-VUE. Doppler shifts vil can be obtained as
where f is the carrier frequency.
3　FRPS algorithm
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3.1　Problem formulation
Practical and hardware-efficient scenarios are considered in which the PSs have finite-resolution to reduce the power consumption and complexity. Within this constraint, the mmWave massive MIMO hybrid precoding algorithms for V2V systems to maximize SE are designed. When Gaussian signals are conveyed via mmWave MIMO channels, the SE can be achieved as
Given finite-resolution PSs and subject to a total power constraint at the Tx, the problem of hybrid precoding design can be articulated as
where A is a feasible set of simulated beamforming matrices at the transmitting and receiving ends. The constraint A is defined by the finite-resolution PSs. To solve Eq. (7), a joint optimization over the four matrix variablesPD, PRF, QD, and QRF is required. The data stream first passes through the digital precoder PD and then through the analog precoder PRF to get the transmission signal x = PRFPDs. After the receiver sends the signal through the channel, it passes through the analog combiner QRF and the digital combiner QD, and finally gets the received signal y. The design of PRF incorporates finite-resolution PSs, which restricts their functionality to phase alteration of signals. The values of each PS are discretized into a finite set, as explained in the next section.
Optimizing Eq. (7) is challenging due to the simultaneous refinement of four matrices, coupled with the non-convex limitations inherent in the RF precoder and combiner. The problems mentioned can be decomposed into two separate optimization problems:the precoding and decoding problems. This decomposition simplifies the design process for the joint hybrid precoding and decoding. While hybrid precoding incorporates extra power constraints, its mathematical formulations share a resemblance with those utilized in decoding problems. Therefore, the design of the precoder will be highlighted.
The hybrid precoding design problem formulation is given by
where PD and PRF are the digital and analog precoders to be optimized. Popt stands for the optimal fully digital precoder. It consists of the first Ns columns from both the matrices V and U, where V and U is obtained by singular value decomposition (SVD) of the channel, that is,H =U ∑ VH. However, Eq. (8) that is no simple closed-form solutions exist because of the coupled PRF and PD as well as the non-convex constraint are set with finite-resolution PSs.
It has been proved that approximate minimization of the objective function in Eq. (8) can maximize the SE[17]. In order to obtain better SE and EE, the digital and analog precoder are jointly designed, and the discrete iterative optimization idea is used to solve the analog and digital precoder alternately. In addition, the influence of discrete phase values should also be considered during optimization.
3.2　FRPS algorithm
It first considers designing the digital precoding matrix PD. Thus Eq. (8) can be reformulated as
Eq. (9) are known to have least squares solutions. Thus, PD can be obtained as
Next, the design of the analog precoder will be highlighted. Applying a constraint to a digital precoding matrix, that is, the columns of the PD should be mutually orthogonal, i. e. , = = , where PB is a unitary matrix with the same dimension as PD, β is a constraint variable. This is inspired by the unconstrained precoding solution. Orthogonal constraints are pivotal in the design of analog precoders, as they can substantially streamline the complexity. By replacing PD with βPB in Eq. (8), the objective function can be further written as
when , the objective function has the minimum value, and can be given by
It should be noted that has the following upper bound
whereis the SVD of and the equality holds when , i. e. , PB is a square matrix. Hence, Eq. (12) can be written as, where PB and PRF are coupled, and it is very complicated to optimize the two variables jointly. In order to rid PRF of the product of PB, the following transformation is performed.
Let's add the constant term to - and multiply it by a positive constant term to get ( - + (2 ·) - +1/2) ×2 . Then it have
Finally, the upper bound is the objective function. To satisfy the power constraint in Eq. (8), it normalize PD by a factor of i. e. ,. ThefollowingTheorem1 will reveal the effect of this normalization.
Theorem 1 Suppose the Euclidean distance before normalization is ‖Popt -PRFPD‖F ≤δ, δ represents the Euclidean distance, and after normalization it becomes .
Proof Define the normalization factor /‖PRFPD‖F = 1/ψ and thus ‖PRFPD‖F = = ψ‖Popt‖F.
By norm inequality, it can be obtained that
Because ‖Popt -PRFPD‖F ≤ δ and‖Popt -PRFPD‖F ≥ |1 - ψ | ‖Popt‖F, it can be obtained ‖Popt‖F≤δ/|ψ-1|.
When ψ≠1, which indicates ‖Popt -PRFPD‖F≠ 0,‖Popt -PRFPD‖F can be rearranged as
Proof end
Therefore, it has been proved that as long as the Euclidean distance between the optimal digital precoder and the hybrid precoders is sufficiently small when ignoring the power constraint in Eq. (8), the normalization step will also achieve a small distance to the optimal digital precoder. With Eq. (14) serving as the objective function and the power constraint being temporarily lifted, the design issue for the hybrid precoder is redefined as
The value of each PS is quantized to a set with a finite number of elements 2B , which is expressed as B ={bΔ|b =0,1,2,... ,2B-1}. B is the number of bits of resolution of PSs and Δ = (2π)/2B is the quantization step size for a uniform quantizer, because of the practical implementation constraints of variable PSs. Accordingly, the elements of the analog precoder PRF are also limited. It denotes the constraint set of the analog precoder as PRFm,n∈A ={exp(j2πb/2B)|b = 0,1,2,... ,2B -1}. Obviously, a larger B will bring more finer resolution for the PSs and better performance, but it will also lead to higher hardware complexity and power consumption.
The objective function in Eq. (17) significantly simplifies the design of the analog precoder. The matrix PRF gets rid of the product form with PD, a closed-form solution can be obtained as
where arg PRF generates a matrix containing the phases of the entries of PRF. Thus, it shows that the phases of PRF can be extracted from the phases of an equivalent precoder . The FRPS algorithm for solving Eq. (17) is presented in Algorithm 1.
Algorithm 1 FRPS algorithm for solving Eq. (17)
In the FRPS algorithm, the dimension of the analog precoder is significantly higher than that of the digital precoder, leading to the complexity of the algorithm being chiefly generated by the analog part. During every cycle of the algorithm, the update of the analog precoder is executed through a phase extraction operation on the matrix , where the dimension of matrix is , and the value of the analog precoder is quantized in a finite set of elements. The algorithm has low complexity. The algorithm calculates PD = βPB in the last step, but this digital precoder should be normalized immediately. Therefore, the PB is normalized directly to satisfy the power constraint without knowing β. By looking at Eq. (6), it can be seen that the SE is not affected by β multiplied with QD. This is because both the received signals and noise have an effect on the QD, and thus the signal-to-noise ratio (SNR) is unaffected by the constant factor β. Besides, not having to calculate the constant β can result in a more streamlined algorithm.
To assess the tradeoff between the performance of the RF chain and the RF complexity in practice, based on the energy consumption model[29], it can specify the EEηas
where Ptotal is the total energy consumption, R is the SE of the system, Pt is the transmitted energy, pRF is the energy consumed by RF chain, pPS is the energy consumed by PS. The number of PSs MPS can be expressed as .
4　Simulation results and analysis
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In this section, the simulation results related to SE and EE of the FRPS algorithm are presented. To visually assess the performance of the FRPS algorithm, it is contrasted against the OMP [17] and phase extraction alternating minimization (PE-AltMin) hybrid algorithms[18]. Both OMP and PE-AltMin algorithms assume an infinite-resolution PS. Additionally, analog and fully digital precoding are also included in the simulation results for ease of comparison.
The parameters for the simulation are outlined below. Every t-VUE is equipped with MT =144 antenna elements. Every r-VUE is equipped with MR = 36 antenna elements. The antenna elements in the UPA are separated by a half wavelength distance i. e. , d = λ/2. A carrier frequency of 28 GHz is designated. Assume that r-VUE and t-VUE are moving at the same speed, vRx =vTx =v =10 m/s. The distance D between the two VUEs is 50 m. The channel matrix can be obtained based on the channel model described in Sect. 2. Clusters count is designated as NC =5, and the rays count within a single cluster is NR = 10[18]. Clusters are endowed with equal power, i. e. , = 1. The AoA and AoD are assumed to follow the uniform distribution within [ - π, π] and angular spread of π/18. The simulation encompasses 5 000 cycles in total.
It is set that the transmits energy Pt =1 W[30], the energy consumed by RF chain pRF = 250 mW[31] and SNR is 10 dB[32]. As the resolution of the PSs decreases, so does the power consumption. Based on the power linear decline model, the power consumption is 24 mW, 18 mW, 12 mW, and 6 mW for B =4 bit, B =3 bit, B =2 bit, and B =1 bit, respectively[32].
Fig. 2 depicts the influence of the PS resolution on SE. The assumption is made that the quantity of RF chains matches the quantity of data streams, i. e. , . This represents the most adverse scenario, given that the count of RF chains is constrained to not fall below the levels stipulated in Sect. 2. It can be seen that with the PS resolution increasing, the SE will improve evidently within B = 3 bit, which proves that the use of FRPS algorithm is beneficial for hybrid precoding schemes. It is observable that an augmentation in the quantity of transmitting antennas results in an enhancement of SE.It can be concluded that 3 bit resolution PS is sufficient for V2V systems using the FRPS algorithm. Compared with high-resolution PS, it offers a more cost-effective and less complex alternative for utilization.
Fig. 3 depicts the convergence characteristics of the FRPS algorithm. Assume that for the worst case.
The PS resolution B = 3 bit are considered in Fig. 3(a). From Fig. 3(a), it becomes clear that no matter how the number of antennas changes, the algorithm converges when the number of iterations is 2. As illustrated in Fig. 3(b), the algorithm converges after 5 iterations with 1 bit resolution PS, after 3 iterations with 2 bit resolution PS and after 2 iterations with 3 bit resolution PS. This indicates that the FRPS algorithm has the capacity for swift convergence, which is a highly favorable property.
Fig. 4 depicts the SE against the SNR for different algorithms. The data stream Ns =2 and =4 is considered. It can be seen that the SE of the FRPS algorithm with 3 bit resolution PSs is close to that of PE-AltMin algorithm with infinite-resolution PSs and even exceeds that of the OMP algorithm with infinite-resolution PSs. The algorithm is much better than the analog precoding algorithm. This is because the analog precoding can only control the signal phase, which brings performance loss. Additionally, it is noted that the FRPS algorithm secures over 90% of the SE of the fully digital precoding algorithm. Though fully digital precoding has the highest SE, it involves higher computational complexity and hardware requirements. It can be concluded that the FRPS algorithm is capable of enhancing the SE with smaller PSs resolution and lower computational complexity. This suggests that it can have greater precision close the performance of the existing infinite-resolution PSs algorithm, which can greatly save the hardware cost.
Fig. 5 depicts the SE versus the number of transmitting antennas. It is assumed that the data stream Ns =2 and . The Rx is equipped with MR = 36 antennas. Notably, as the number of antennas for transmission rises, there is a corresponding enhancement in SE. This is because the amplification of the quantities of antennas can provide more antenna diversity and greater interference immunity. Furthermore, the results indicate that the performance of the developed algorithm is close to that of the PE-AltMin algorithm and fully digital precoding strategies. It is gradually better than that of the OMP algorithm when the number of antennas is more than 100. It means that the performance of the FRPS algorithm is excellent as the number of transmitting antennas rises. The analysis shows that the SE of the analog precoding is lowest and grows slowly with the increase of the number of antennas, and the SE is much lower than that of other precoding algorithms. It can be concluded that the use of analog precoding is not feasible in the V2V system.
Fig. 6 depicts the influence of the PS's resolution on EE.
It is assumed that as Fig. 3 for the worst case. It can be observed that enhancing the resolution of the PS leads to a reduction in the EE. This means that the resolution of the PS cannot be blindly improved in the actual V2V system design. Additionally, the EE tends to decline with an escalating number of antennas utilized for transmission.
Fig. 7 depicts the EE versus the number of RF chains. The data stream Ns =2 and . It can be observed that the EE of the FRPS algorithm outperforms that of OMP, PE-AltMin, and fully digital precoding algorithms. Furthermore, it becomes clear that the EE of the FRPS algorithm when the RF chain is less than 3 is better than that of the analog precoding with one RF chain. It can be observed that the EE decreases with the increase of the RF chains with OMP, PE-AltMin, and the FRPS algorithm.
Fig. 8 depicts the EE versus the SE with different resolution of PSs. The data stream Ns =2 and = =4. It is shown that the V2V system performance improves as the resolution increases. However, although the improvement of resolution can make the phase quantization more accurate, the power consumption grows much faster than the SE with the increase of B, which leads to a sharp decline in EE. It implies that the use of infinite-resolution PSs in V2V system is not practical and causes large energy losses. The results in Fig. 8 further demonstrate the rationality of setting B =3 bit for the FRPS algorithm. When B ＞ 3 bit, the SE will not increase significantly, but the EE will decrease significantly. Furthermore, observations indicate that with the increase of the number of transmitting antennas, the SE will increase but the EE will decrease.
Fig. 9 depicts the EE versus the SE with different numbers of RF chains. The data stream Ns = = . It can be seen that with the increase of RF chains, the SE of the V2V system increases, but the EE decreases. This is because the hardware and software complexity will increase with increasing the number of RF chains, which may lead to higher design and maintenance costs, and require more energy consumption.
By increasing RF chains, the signal can be controlled more accurately. The EE of the FRPS algorithm when is even better than the PE-AltMin algorithm when . This means that when the same energy is consumed, the FRPS algorithm can achieve greater SE by setting more RF chains to make up for the loss caused by the insufficient resolution of the PSs. From the perspective of both EE and SE, the FRPS algorithm is superior to other algorithms.
Fig. 10 depicts the EE versus the SE with different numbers of transmitting antennas. The data stream Ns =2 and . It is observable that with increasing the number of transmitting antennas, the SE increases, but the EE decreases. It is clear that the FRPS algorithm performs excellently in terms of EE, approaching the lowest energy consumption with analog precoding regardless of the number of transmitting antennas.
5　Conclusions
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In this paper, a two-stage iterative hybrid precoding algorithm, FRPS, was proposed for the mmWave massive MIMO V2V system. The digital and analog precoders are jointly designed, and the discrete iterative optimization idea is used to solve the analog and digital precoders alternately. Numerical simulations confirm the effectiveness of the algorithm. As the data indicated, the FRPS algorithm demonstrates fast convergence and also approaches the SE of fully digital precoding with only 3 bit resolution PS. Moreover, the SE delivered by the FRPS algorithm outperforms the OMP algorithm and closely rivals the performance of the PE-AltMin algorithm. The EE of the FRPS algorithm outperforms the other infinite-resolution PS hybrid precoding algorithm and fully digital precoding algorithm.
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doi: 10.19682/j.cnki.1005-8885.2025.0017
  • Online Date：2026-04-17
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    1College of Electronic and Information Engineering, Hebei University, Baoding 071002, China
    2Hebei Key Laboratory of Power Internet of Things Technology, North China Electric Power University, Baoding 071003, China
    3School of Electrical and Electronic Engineering, North China Electric Power University, Baoding 071003, China

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Corresponding author: Rong Zhanyi, E-mail:
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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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