CN116094556A - Spatial multiplexing method based on IRS auxiliary terahertz MIMO communication system - Google Patents

Abstract

The invention relates to a space multiplexing method based on an IRS-assisted terahertz MIMO communication system, and belongs to the technical field of communication. The method includes the following steps: the IRS multi-partition auxiliary transceiver terminal multi-subarray terahertz MIMO communication system architecture is proposed; under the proposed architecture, a channel model is established based on the Kronecker product; according to the principle of spectrum efficiency maximization, a Non-convex objective function with multi-variable coupling and non-convex constraints; decoupling the optimization problem into two easy-to-solve sub-problems, namely the IRS reflection coefficient matrix design problem and the receiving/transmitting hybrid precoding/combination matrix design problem; based on Riemann flow The shape optimization algorithm is used to calculate the IRS reflection coefficient matrix; based on mathematical derivation, the closed-form solution of the hybrid precoding matrix/combination matrix is obtained.

Description

Spatial multiplexing method based on IRS-assisted terahertz MIMO communication system

Technical Field

The invention belongs to the technical field of communication and relates to a spatial multiplexing method based on an IRS-assisted terahertz MIMO communication system.

Background Art

In recent years, the construction of 5G is in full swing around the world. At the same time, academia and industry have explored the new model of 6G, and made preliminary ideas and research on the vision, requirements, scenarios, key technologies, system architecture and performance indicators of 6G networks, and put forward the development goals and overall vision of full coverage, full spectrum, full application and strong security. Compared with 5G, the performance indicators of 6G, such as peak rate, connection density and spectrum efficiency, have been improved by 10 to 100 times. Terahertz (THz) communication is favored by the next generation of wireless communication systems because of its ultra-large bandwidth and high data transmission rate requirements. However, the ultra-high path loss in the THz band limits the communication distance of THz communication. For this reason, it is often combined with massive multiple input multiple output (MIMO) technology, using the high array gain generated by massive MIMO to compensate for the path loss, while providing multiplexing gain to further improve the spectrum efficiency of the system. However, the high hardware cost and high energy consumption of massive MIMO technology have brought challenges to the actual deployment of the network.

The emerging IRS (Intelligent Reflecting Surface, IRS) technology provides a turning point for effectively solving network deployment problems. Due to its advantages such as low cost, easy deployment, and active and intelligent regulation of the wireless propagation environment, IRS has been included in the key enabling technology of the next generation of wireless communications. Specifically, IRS is a reconfigurable plane composed of a large number of low-cost passive reflective elements. Each reflective element independently adjusts the phase shift and amplitude of the incident electromagnetic wave in a programmable manner, thereby collaboratively realizing reflection beamforming and reconfiguring the propagation environment. Therefore, the use of IRS in existing wireless communication systems can create a good propagation environment and provide more degrees of freedom for optimization. By reasonably regulating the physical properties of each reflective unit, the electromagnetic wave signal reflected by the IRS can form a reflection beamforming, thereby gathering the energy of the reflected signal and pointing it to the receiving end, improving the received signal strength, and improving the system capacity.

At present, the application of IRS in wireless communication has been widely studied. Among many key technology studies, how to jointly optimize the RIS reflection coefficient and the transmitter beamforming matrix to maximize the IRS performance gain is a key issue. In this regard, most studies solve the IRS reflection coefficient matrix and the hybrid precoding matrix of the transceiver through alternating optimization, or the alternating optimization precoding design of the transceiver, or the alternating optimization design of the inner and outer layers of digital precoding and analog precoding, or the alternating optimization design of the IRS reflection phase shift matrix and the hybrid precoding matrix. However, the alternating optimization method has the problem of high computational complexity, which has given rise to improved optimization algorithms such as block coordinate descent algorithm, semidefinite relaxation algorithm, truncated channel matrix singular value decomposition method, etc., to achieve a balance between system performance and computational complexity. However, the above methods are all based on the optimization algorithm. The inherent half-wavelength antenna array architecture of the system and the consideration of plane wave assumption in information transmission make the spatial multiplexing gain of the system limited by the number of resolvable paths. Especially in high-frequency band communications such as THz communications, the channels have extremely high propagation attenuation and scattering losses, and the channels are sparse. Relying on the spatial multiplexing method of the traditional architecture, there is limited room for improving the algorithm to obtain spectrum efficiency gains, and the spectrum efficiency of the IRS-assisted TH-MIMO system has encountered a bottleneck. Therefore, the proposal of a new architecture is a major breakthrough in improving the spatial multiplexing gain and system spectrum efficiency of the IRS-assisted TH-MIMO system.

This paper proposes an IRS multi-partition assisted transceiver multi-subarray terahertz MIMO communication system architecture for IRS-assisted THz-MIMO point-to-point communication systems; under the proposed architecture, a channel model is established based on the Kronecker product; with the goal of maximizing the spectral efficiency of the system, a non-convex optimization function is constructed, and the original problem is decoupled into two sub-problems for solution by taking advantage of the uncoupled constraints of the optimization function. The difference is that the traditional plane wave assumption is no longer considered in this paper, but spherical wave propagation is considered between different sub-arrays of the transceiver and between different IRS groups, while optimizing the IRS reflection beamforming, the hybrid precoding and combination matrix of the transceiver are derived.

Summary of the invention

In view of this, an object of the present invention is to provide a spatial multiplexing method based on an IRS-assisted terahertz MIMO communication system.

In order to achieve the above object, the present invention provides the following technical solutions:

A spatial multiplexing method based on an IRS-assisted terahertz MIMO communication system, the method comprising the following steps:

Step 1: The IRS multi-partition auxiliary transceiver multi-subarray terahertz MIMO communication system architecture is proposed;

Step 2: Under the proposed architecture, a channel model is established based on the Kronecker product;

Step 3: According to the principle of maximizing spectrum efficiency, a non-convex objective function containing multi-variable coupling and non-convex constraints is constructed under the proposed architecture;

Step 4: Decouple the optimization problem into two easily solvable sub-problems, namely, the IRS reflection coefficient matrix design problem and the hybrid precoding/combination matrix design problem at the receiving/transmitting end;

Step 5: Calculate the IRS reflection coefficient matrix based on the Riemann manifold optimization algorithm;

Step 6: Based on mathematical derivation, the closed-form solution of the hybrid precoding matrix/combination matrix is obtained.

Optionally, in the step one, the IRS multi-partition assisted transceiver multi-subarray terahertz MIMO communication system architecture proposes that, for the IRS-assisted terahertz MIMO system, it is assumed that the line-of-sight communication link between the transmitter and the receiver is blocked by an obstacle, and it is necessary to rely on the IRS to establish an effective communication link; in order to obtain a richer spatial multiplexing gain, a wide-interval multi-subarray hybrid precoding structure is adopted at the transceiver, and a corresponding wide-interval multi-partition IRS architecture is designed.

Optionally, in the step 2, under the proposed architecture, based on the Kronecker product, the channel model established is combined with the wide-spaced multi-subarray WSMS architecture channel model and the IRS cascade channel model to obtain a virtual line-of-sight communication channel constructed by the transmitter and the receiver through the IRS, which is expressed as

H＝H _r ΦH _t

发送端与IRS之间的信道H_t，IRS与接收端之间的信道H_r表示为In the formula

The channel _Ht between the transmitter and IRS and the channel _Hr between the IRS and the receiver are expressed as

Optionally, in the step three, according to the principle of maximizing spectrum efficiency, a non-convex objective function containing multivariable coupling and non-convex constraints is constructed under the proposed architecture, and the limitations of the limited scattering path on the multiplexing gain are broken by deploying antenna arrays at both ends of the transmission and reception and elements on the IRS, breaking through the spectrum efficiency bottleneck in the existing IRS-assisted terahertz MIMO communication system, aiming to maximize the system spectrum efficiency by jointly optimizing the reflection coefficient matrix on the IRS, the hybrid precoding matrix at the transmitting end, and the hybrid merging matrix at the receiving end; the system spectrum efficiency is

The optimization problem of maximizing the system spectrum efficiency is stated as

Optionally, in step 4, the optimization problem is decoupled into two easily solvable sub-problems, namely, the IRS reflection coefficient matrix design problem and the hybrid precoding/combination matrix design problem of the transmitting/receiving end. When solving each problem, we assume that the channel state information is completely known, and focus on the joint beamforming on the transmitting/receiving end and the IRS under the proposed new architecture; first, assuming that the hybrid precoding matrix of the transmitting/receiving end is fully digital, with the goal of maximizing the spectrum efficiency of the system, the reflection coefficient matrix on the IRS is optimized, and the first optimization sub-problem obtained is

φ∈[0,2π)

Substitute the obtained IRS reflection coefficient matrix to optimize the hybrid precoding matrix (merging matrix) at the transmitting (receiving) end, and then get the second optimization sub-problem

Optionally, in step 5, the IRS reflection coefficient matrix is calculated based on the Riemann manifold optimization algorithm. Since P1 assumes that the precoding and combination matrices of the transceiver are in the optimal form of full digital, by further analyzing the structure of the cascaded channel matrix, the optimization problem of P1 is simplified to the following form:

φ∈[0,2π)

和

分别表示

和

的第k行和第k列，

和

分别表示对x向上和向下取整，

表示元素全为1的行向量，1_K∈C^K×1表示元素全为1的列向量；令

in

and

Respectively

and

The kth row and kth column of

and

They represent rounding x up and down respectively.

represents a row vector whose elements are all 1, 1 _K ∈ C ^K×1 represents a column vector whose elements are all 1; let

则优化问题式P1重新表述为

Then the optimization problem P1 can be reformulated as

φ∈[0,2π)

The feasible search space of the transformed optimization problem is regarded as the product of _{Nirs_tot} complex circles, that is:

When searching for the optimal phase shift on the manifold M, the constant modulus constraint of the IRS reflection coefficient is always satisfied, P1 is converted into an unconstrained form, and the gradient descent algorithm is used to solve it.

Optionally, in step 6, based on mathematical derivation, a closed-form solution of the hybrid precoding matrix/combination matrix is obtained. First, the cascaded channel is decomposed by SVD.

Q是级联信道矩阵H的秩；通过进一步剖析级联信道矩阵的结构，简化P2的优化问题，将级联信道矩阵重写为如下形式Where U is a unitary matrix of N _{r_tot} ×Q, Σ is a Q×Q diagonal matrix whose diagonal elements are the singular values of the concatenated channels, and V is a unitary matrix of N _{t_tot} ×Q.

Q is the rank of the cascade channel matrix H. By further analyzing the structure of the cascade channel matrix and simplifying the optimization problem of P2, the cascade channel matrix can be rewritten as follows:

结合H的SVD分解，得到发送端混合预编码矩阵的闭式解in,

Combined with the SVD decomposition of H, the closed-form solution of the hybrid precoding matrix at the transmitter is obtained:

表示右奇异矩阵的前N_s列，

的归一化注水功率分配矩阵，

表示第i条数据流分配的功率，且i＝1,2,L,N_s，ε是注水高度，

将级联信道矩阵重写为：in,

represents the first N _s columns of the right singular matrix,

The normalized water injection power allocation matrix is:

represents the power allocated to the ith data stream, and i＝1,2,L,N _s , ε is the water injection height,

Rewrite the cascaded channel matrix as:

结合H的SVD分解，得到接收端混合组合码矩阵的闭式解in,

Combined with the SVD decomposition of H, the closed-form solution of the mixed combination code matrix at the receiving end is obtained

表示左奇异矩阵的前N_s列。in,

Represents the first _Ns columns of the left singular matrix.

The beneficial effects of the present invention are:

1) An IRS multi-partition assisted transceiver multi-subarray terahertz MIMO communication system architecture is proposed. Since the spacing between the IRS and the transceiver subarrays is wide and the correlation between the subarrays is low, the resolvable phase difference brought to each subarray by spherical wave transmission is used to obtain the inter-path multiplexing gain and the intra-path multiplexing gain at the same time, breaking the limitation of the spatial multiplexing gain in the traditional architecture due to the sparsity of the high-frequency communication channel;

2) Under the IRS multi-partition assisted transceiver multi-subarray terahertz MIMO communication system architecture, a channel model based on Kronecker product was established by jointly analyzing the communication channel model under the wide subarray architecture and the cascade channel model of IRS assisted communication;

3) With the goal of maximizing the spectral efficiency of the system, a non-convex optimization function was constructed. The original problem was decoupled into two sub-problems by utilizing the uncoupled constraints of the optimization function. The IRS reflection coefficient matrix was calculated by dissecting the channel structure and using the Riemann manifold optimization algorithm. The closed-form solution of the hybrid precoding matrix/combination matrix was obtained through mathematical deduction, achieving a good compromise between computational complexity and overall spectral efficiency.

Other advantages, objectives and features of the present invention will be described in the following description to some extent, and to some extent, will be obvious to those skilled in the art based on the following examination and study, or can be taught from the practice of the present invention. The objectives and other advantages of the present invention can be realized and obtained through the following description.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to make the purpose, technical solutions and advantages of the present invention more clear, the present invention will be described in detail below in conjunction with the accompanying drawings, wherein:

FIG1 is a design process of a spatial multiplexing scheme based on an IRS-assisted terahertz MIMO communication system;

FIG2 is a model diagram of the IRS multi-partition auxiliary transceiver multi-subarray terahertz MIMO communication system;

Figure 3 is a geometric diagram of Riemannian manifold optimization.

DETAILED DESCRIPTION

The following describes the embodiments of the present invention by specific examples, and those skilled in the art can easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments only illustrate the basic concept of the present invention in a schematic manner, and the following embodiments and the features in the embodiments can be combined with each other without conflict.

Among them, the drawings are only used for illustrative explanations, and they only represent schematic diagrams rather than actual pictures, and should not be understood as limitations on the present invention. In order to better illustrate the embodiments of the present invention, some parts of the drawings may be omitted, enlarged or reduced, and do not represent the size of actual products. For those skilled in the art, it is understandable that some well-known structures and their descriptions in the drawings may be omitted.

The same or similar numbers in the drawings of the embodiments of the present invention correspond to the same or similar parts; in the description of the present invention, it should be understood that if the terms "upper", "lower", "left", "right", "front", "back" and the like indicate directions or positional relationships, they are based on the directions or positional relationships shown in the drawings, which are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific direction, be constructed and operated in a specific direction. Therefore, the terms describing the positional relationship in the drawings are only used for illustrative purposes and cannot be understood as limiting the present invention. For ordinary technicians in this field, the specific meanings of the above terms can be understood according to specific circumstances.

As shown in FIG1 , the present invention provides a spatial multiplexing scheme design based on an IRS-assisted terahertz MIMO communication system.

FIG2 is a system model diagram of the present invention, which will be described below in conjunction with the accompanying drawings:

The IRS-assisted terahertz MIMO communication system model considered in the present invention is shown in Figure 2. The system consists of three parts: a transmitter, an IRS, and a receiver. The transmitter includes N _{t_tot} RF chains and N _{t_tot} antennas, which evenly divide the transmitter into K _t subarrays with a uniform interval of d _{wid_t} , and each subarray is configured with N _{RF_t} RF chains and N _t antennas with a spacing of d=λ/2; the IRS includes N _{irs_tot} reflective elements, which are evenly divided into K _irs groups with a group spacing of d _{wid_irs} , and each group of IRS includes N _irs reflective elements with a uniform spacing of d=λ/2; the receiver includes N _{r_tot} RF chains and N _{r_tot} antennas, which evenly divide the receiver into K _r subarrays with a uniform interval of d _{wid_r} , and each subarray is configured with N _{RF_r} RF chains and N _r antennas with a spacing of d=λ/2.

且有

E表示期望，(A)^H表示矩阵的共轭转置，

为N_s×N_s的单位矩阵。其中，传输数据流数、收发端RF链数、收发端天线数存在如下关系：N_s≤N_{RFt_tot}≤N_{t_tot}，N_s≤N_{RFr_tot}≤N_{r_tot}，1≤N_{RF_t}≤N_t，1≤N_{RF_r}≤N_r。发射信号s首先经过数字预编码器

通过RF链路映射到射频域，再经过模拟预编码器

的相移网络，通过发送端天线辐射后得到发送信号为The transmitter sends N _S parallel data streams, expressed as

And there is

E represents expectation, (A) ^H represents the conjugate transpose of the matrix,

is the identity matrix of N _s ×N _s . Among them, the number of transmission data streams, the number of RF chains at the transmitting and receiving ends, and the number of antennas at the transmitting and receiving ends have the following relationship: N _s ≤N _{RFt_tot} ≤N _{t_tot} , N _s ≤N _{RFr_tot} ≤N _{r_tot} , 1≤N _{RF_t} ≤N _t , 1≤N _{RF_r} ≤N _r . The transmission signal s first passes through the digital precoder

Mapped to the RF domain through the RF link, and then passed through the analog precoder

The phase shift network is radiated by the transmitting antenna to obtain the transmitted signal:

每个子阵的模拟预编码矩阵

θ_i,n为子阵列F_RF,i的第n列向量，

n＝1,2,3,L,N_{RF_t}，模拟预编码矩阵中的非零元素均满足恒模约束，即

混合预编码器满足功率约束

其中||·||_F表示Frobenius范数。Where ρ represents the signal transmission power. Since the RF chains between the subarrays are independent of each other, the analog precoding matrix F _RF is a block diagonal structure.

The simulated precoding matrix for each subarray

θ _i,n is the nth column vector of subarray F _RF,i ,

n＝1,2,3,L,N _{RF_t} , the non-zero elements in the analog precoding matrix all satisfy the constant modulus constraint, that is,

Hybrid precoder meets power constraints

where ||·|| _F denotes the Frobenius norm.

IRS与接收端之间的信道为

IRS上的相移矩阵为

在IRS辅助通信中，当收发端的子阵列之间的间距以及IRS的组间距较大时，此时平面波传输近似不再适用，需要考虑球形波。因此，结合宽间距多子阵列(WSMS)架构信道模型和IRS级联信道模型，得出发送端与接收端通过IRS构建的虚拟视距通信信道可以表示为The channel matrix between the transmitter and IRS is defined as

The channel between IRS and the receiver is

The phase shift matrix on IRS is

In IRS-assisted communication, when the spacing between the subarrays of the transmitter and receiver and the group spacing of the IRS are large, the plane wave transmission approximation is no longer applicable, and spherical waves need to be considered. Therefore, combining the wide-spaced multi-subarray (WSMS) architecture channel model and the IRS cascade channel model, it is concluded that the virtual line-of-sight communication channel constructed by the transmitter and the receiver through the IRS can be expressed as

H＝H _r ΦH _t

in,

表示克罗内克积，

和

是信道的复增益，L_t和L_r分别表示发送端与IRS之间、IRS与接收端之间的传播路径数。

和

分别表示信号到达(离开)IRS的第l_t(l_r)条路径的方位角和俯仰角，

为IRS上均匀平面子阵列响应向量，

和

分别表示信号离开(到达)发送(接收)端第l_t(l_r)条路径的方位角和俯仰角，

为发送(接收)端均匀平面子阵列响应向量。

和

分别表示球面波传播下发送端与IRS之间的第l_t条路径上子阵列之间的复相移矩阵和IRS与接收端之间的第l_r条路径上子阵列之间的复相移矩阵，且有

其中

表示发送端的第k_t个子阵列在路径l_t方向上与IRS上第k_irs组之间的距离，

表示接收端的第k_r个子阵列在路径l_r方向上与IRS上第k_irs组之间的距离。in,

represents the Kronecker product,

and

is the complex gain of the channel, _Lt and _Lr represent the number of propagation paths between the transmitter and IRS and between the IRS and the receiver, respectively.

and

denote the azimuth and elevation angles of the l _t (l _r )th path of the signal arriving at (leaving) the IRS,

is the uniform plane subarray response vector on IRS,

and

They represent the azimuth and elevation angles of the l _t (l _r )th path of the signal leaving (arriving) the transmitting (receiving) end, respectively.

is the uniform plane subarray response vector of the transmitting (receiving) end.

and

denote the complex phase shift matrix between subarrays on the lt _- th path between the transmitter and the IRS and the complex phase shift matrix between subarrays on the lr _- th path between the IRS and the receiver under spherical wave propagation, and

in

represents the distance between the kt _-th subarray at the transmitter and the kirs- _th group on the IRS in the direction of path _lt ,

represents the distance between the k _r th subarray at the receiving end and the k _{irs th} group on the IRS in the direction of path l _r .

和

分别表示x轴上包含

个阵元的均匀线性阵列(ULA)的阵列响应向量和表示z轴上包含

个阵元的均匀线性阵列(ULA)的阵列响应向量。于接收端子阵列而言，有

个阵元，则接收端的阵列响应向量in,

and

Respectively represent the x-axis contains

The array response vector of a uniform linear array (ULA) with 10 elements and 1000 elements represents the z-axis containing

The array response vector of a uniform linear array (ULA) with 10 elements. For the receiving terminal array, there is

array elements, then the array response vector at the receiving end is

可以表示为In addition, the IRS normalized subarray response vector

It can be expressed as

同样地，通过变换

中的上下标，即可得到where x and y represent the indices of the elements in the subarray on the IRS, and

Similarly, by transforming

The superscripts and subscripts in , we can get

The transmitted signal x reaches the IRS through the channel _Ht . Under the control of the FPGA controller, the IRS applies a phase shift Φ to the received signal, and the IRS reflected signal reaches the receiving end through _Hr . Therefore, the signal received by the receiving end is

是信道中的加性高斯白噪声，且n:

由于IRS上各个元件的反射系数是相互独立，所以IRS上各组元件的反射系数之间也是相互独立的，则IRS上的相移矩阵是块对角矩阵，

每个IRS组的反射系数矩阵

各个元件的反射系数为

k＝1,2,L,K_irs，l＝1,2,L,N_irs，γ_k,l和φ_k,l分别是第k组IRS上第l元件的反射幅值和相移，一般地无源IRS的幅值γ_k,l＝1，反射相移φ_k,l∈[0,2π)。接收端天线接收到的信号经过模拟组合器

和数字组合器

后得到的信号为in,

is the additive white Gaussian noise in the channel, and n:

Since the reflection coefficients of each element on the IRS are independent of each other, the reflection coefficients of each group of elements on the IRS are also independent of each other. Therefore, the phase shift matrix on the IRS is a block diagonal matrix.

The reflection coefficient matrix for each IRS group

The reflection coefficient of each element is

k＝1,2,L,K _irs ，l＝1,2,L,N _irs ，γ _k,l and φ _k,l are respectively the reflection amplitude and phase shift of the lth element on the kth group of IRS. Generally, the amplitude of passive IRS is γ _k,l ＝1, and the reflection phase shift is φ _k,l ∈[0,2π). The signal received by the receiving antenna is passed through the analog combiner

and digital combiners

The signal obtained is

Similar to the analog precoder at the transmitter, _WRF is also block diagonalized and satisfies the constant modulus constraint, i.e.

Under the proposed system framework, the main goal is to maximize the system's spectral efficiency by jointly optimizing the hybrid precoding matrix/combined matrix at the receiving/transmitting end and the reflected phase matrix at the IRS end. First, the system spectral efficiency is

表示噪声的协方差矩阵。因此，最大化系统频谱效率的优化问题可以表述为In the formula,

represents the covariance matrix of the noise. Therefore, the optimization problem of maximizing the system spectral efficiency can be expressed as

φ∈[0,2π)

The optimization problem is decoupled into two easily solvable sub-problems, namely, the IRS reflection coefficient matrix design problem and the hybrid precoding/combination matrix design problem at the transmitter/receiver. When solving each problem, we assume that the channel state information is completely known and focus on the joint beamforming at the transmitter/receiver and IRS under the proposed new architecture. Specifically, we first assume that the hybrid precoding matrix (combination matrix) at the transmitter/receiver is fully digital, and optimize the reflection coefficient matrix on the IRS with the goal of maximizing the system's spectral efficiency. The first optimization sub-problem is

φ∈[0,2π)

Substitute the obtained IRS reflection coefficient matrix to optimize the hybrid precoding matrix (merging matrix) at the transmitting (receiving) end, and then get the second optimization sub-problem

FIG3 is a schematic diagram of the geometric interpretation of the Riemann manifold optimization method. The following is an explanation in conjunction with the accompanying drawings:

Based on the Riemann manifold optimization algorithm, the IRS reflection coefficient matrix is calculated. Since in P1, it is assumed that the precoding and combination matrices of the transmitting and receiving ends are in the optimal form of full digital, by further analyzing the structure of the cascade channel matrix, the optimization problem of P1 is simplified and the cascade channel matrix is rewritten as follows:

和

分别表示K_r×K_r和K_t×K_t维的单位矩阵，

且in

and

denote the identity matrices of K _r ×K _r and K _t ×K _t dimensions respectively,

and

当收发端的天线阵列足够大时，A_R和A_T可认为是标准正交矩阵，两个矩阵的列向量分别构成各自的正交集，根据克罗内克积的性质知

和

也是标准的正交矩阵。若合理设计IRS上的反射系数，使得矩阵D主对角线上的元素远大于非主对角线上的元素，则

可以近似看作级联信道矩阵H的SVD分解。因此优化问题P1可以转换为如下形式in

When the antenna arrays at the transmitting and receiving ends are large enough, _AR and _AT can be considered as standard orthogonal matrices. The column vectors of the two matrices constitute their own orthogonal sets. According to the properties of the Kronecker product,

and

It is also a standard orthogonal matrix. If the reflection coefficient on the IRS is properly designed so that the elements on the main diagonal of the matrix D are much larger than the elements on the non-main diagonal, then

It can be approximately regarded as the SVD decomposition of the cascade channel matrix H. Therefore, the optimization problem P1 can be transformed into the following form

φ∈[0,2π)

和

分别表示

和

的第k行和第k列，

和

分别表示对x向上和向下取整，

表示元素全为1的行向量，1_K∈C^K×1表示元素全为1的列向量。令

in

and

Respectively

and

The k-th row and k-th column of

and

They represent rounding x up and down respectively.

represents a row vector whose elements are all 1, and 1 _K ∈ C ^K×1 represents a column vector whose elements are all 1.

则优化问题式P1可以重新表述为

Then the optimization problem P1 can be restated as

φ∈[0,2π)

The feasible search space of the transformed optimization problem can be regarded as the product of _{Nirs_tot} complex circles, that is:

When searching for the optimal phase shift on the manifold M, the constant modulus constraint of the IRS reflection coefficient is always satisfied. Therefore, P1 can be converted into an unconstrained form and solved using the gradient descent algorithm. At this time, the optimized objective function is

In the Riemann manifold, the fastest descent direction of the objective function is the direction associated with the negative Riemann gradient, which can be obtained by Euclidean gradient mapping. Therefore, first, calculate the Euclidean gradient of the objective function f(v) at v ^k

投影到切空间

上，并计算f(v)在v^k处的黎曼梯度Next, use the orthogonal projection operator Proj(.) to transform the Euclidean gradient

Projection to tangent space

and calculate the Riemann gradient of f(v) at v ^k

Then, v ^k is updated along the negative Riemann degree direction according to the step size μ ^k

位于切空间，需要使用收缩算子将更新的点重新映射回流形，以便继续使用负黎曼梯度，进行下一步的更新。

映射到流形上的v^k+1处Where μ ^k represents the Armijo step size.

Located in the tangent space, a contraction operator is needed to remap the updated points back to the manifold so that the negative Riemann gradient can be continued for the next update.

Mapped to v ^k+1 on the manifold

According to the above steps, the optimal solution of the IRS reflection coefficient matrix can be obtained.

After substituting the obtained IRS reflection coefficient matrix into the original optimization problem, the cascade channel is decomposed by SVD

Q是级联信道矩阵H的秩。通过进一步剖析级联信道矩阵的结构，简化P2的优化问题，将级联信道矩阵重写为如下形式Where U is a unitary matrix of N _{r_tot} ×Q, Σ is a Q×Q diagonal matrix whose diagonal elements are the singular values of the concatenated channels, and V is a unitary matrix of N _{t_tot} ×Q.

Q is the rank of the cascade channel matrix H. By further analyzing the structure of the cascade channel matrix and simplifying the optimization problem of P2, the cascade channel matrix can be rewritten as follows:

结合H的SVD分解，得到发送端混合预编码矩阵的闭式解in,

Combined with the SVD decomposition of H, the closed-form solution of the hybrid precoding matrix at the transmitter is obtained:

表示右奇异矩阵的前N_s列，

的归一化注水功率分配矩阵，

表示第i条数据流分配的功率，且i＝1,2,L,N_s，ε是注水高度，

相似地，将级联信道矩阵重写为in,

represents the first N _s columns of the right singular matrix,

The normalized water injection power allocation matrix is:

represents the power allocated to the ith data stream, and i＝1,2,L,N _s , ε is the water injection height,

Similarly, the cascaded channel matrix can be rewritten as

结合H的SVD分解，得到接收端混合组合码矩阵的闭式解in,

Combined with the SVD decomposition of H, the closed-form solution of the mixed combination code matrix at the receiving end is obtained

表示左奇异矩阵的前N_s列。in,

Represents the first _Ns columns of the left singular matrix.

Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solution of the present invention can be modified or replaced by equivalents without departing from the purpose and scope of the technical solution, which should be included in the scope of the claims of the present invention.

Claims (7)

1. A spatial multiplexing method based on an IRS-assisted terahertz MIMO communication system, characterized in that the method comprises the following steps: Step 1: The IRS multi-partition auxiliary transceiver multi-subarray terahertz MIMO communication system architecture is proposed; Step 2: Under the proposed architecture, a channel model is established based on the Kronecker product; Step 3: According to the principle of maximizing spectrum efficiency, a non-convex objective function containing multi-variable coupling and non-convex constraints is constructed under the proposed architecture; Step 4: Decouple the optimization problem into two easily solvable sub-problems, namely, the IRS reflection coefficient matrix design problem and the hybrid precoding/combination matrix design problem at the receiving/transmitting end; Step 5: Calculate the IRS reflection coefficient matrix based on the Riemann manifold optimization algorithm; Step 6: Based on mathematical derivation, the closed-form solution of the hybrid precoding matrix/combination matrix is obtained. 2. According to claim 1, the spatial multiplexing method based on the IRS-assisted terahertz MIMO communication system is characterized in that: in the step 1, the IRS multi-partition assisted transceiver multi-subarray terahertz MIMO communication system architecture proposes that, for the IRS-assisted terahertz MIMO system, it is assumed that the line-of-sight communication link between the transmitter and the receiver is blocked by an obstacle, and it is necessary to rely on the IRS to establish an effective communication link; in order to obtain a richer spatial multiplexing gain, a wide-interval multi-subarray hybrid precoding structure is adopted at the transceiver, and a corresponding wide-interval multi-partition IRS architecture is designed. 3. The spatial multiplexing method based on the IRS-assisted terahertz MIMO communication system according to claim 1 is characterized in that: in the step 2, under the proposed architecture, based on the Kronecker product, the channel model established is combined with the wide-spaced multi-subarray WSMS architecture channel model and the IRS cascade channel model to obtain a virtual line-of-sight communication channel constructed by the transmitter and the receiver through the IRS. H＝H _r ΦH _t In the formula

The channel _Ht between the transmitter and IRS and the channel _Hr between the IRS and the receiver are expressed as

4. The spatial multiplexing method based on the IRS-assisted terahertz MIMO communication system according to claim 1 is characterized in that: in the step 3, according to the principle of maximizing spectrum efficiency, a non-convex objective function containing multivariable coupling and non-convex constraints is constructed under the proposed architecture, and the limitations of the limited scattering path on the multiplexing gain are broken by deploying antenna arrays at both ends of the transmission and reception and elements on the IRS, breaking through the spectrum efficiency bottleneck in the existing IRS-assisted terahertz MIMO communication system, aiming to maximize the system spectrum efficiency by jointly optimizing the reflection coefficient matrix on the IRS, the hybrid precoding matrix at the transmitting end, and the hybrid merging matrix at the receiving end; the system spectrum efficiency is

The optimization problem of maximizing the system spectrum efficiency is stated as

5. The spatial multiplexing method based on the IRS-assisted terahertz MIMO communication system according to claim 1 is characterized in that: in the step 4, the optimization problem is decoupled into two easily solvable sub-problems, namely, the IRS reflection coefficient matrix design problem and the hybrid precoding/combination matrix design problem of the receiving/transmitting end. When solving each problem, we assume that the channel state information is completely known, and focus on studying the joint beamforming on the transceiver and IRS under the proposed new architecture; first, assuming that the hybrid precoding matrix of the transceiver is fully digital, with the goal of maximizing the spectrum efficiency of the system, the reflection coefficient matrix on the IRS is optimized, and the first optimization sub-problem obtained is P1:

φ∈[0,2π) Substitute the obtained IRS reflection coefficient matrix to optimize the hybrid precoding matrix (merging matrix) at the transmitting (receiving) end, and then get the second optimization sub-problem P2:

6. The spatial multiplexing method based on the IRS-assisted terahertz MIMO communication system according to claim 1 is characterized in that: in the step 5, the IRS reflection coefficient matrix is calculated based on the Riemann manifold optimization algorithm. Since the precoding and combination matrices of the transceiver are assumed to be in the optimal form of all-digital in P1, by further analyzing the structure of the cascaded channel matrix, the optimization problem of P1 is simplified to the following form:

φ∈[0,2π) in

and

Respectively

and

The k-th row and k-th column of

and

They represent rounding x up and down respectively.

represents a row vector whose elements are all 1, 1 _K ∈ C ^K×1 represents a column vector whose elements are all 1; let

Then the optimization problem P1 can be reformulated as

φ∈[0,2π) The feasible search space of the transformed optimization problem is regarded as the product of _{Nirs_tot} complex circles, that is:

When searching for the optimal phase shift on the manifold M, the constant modulus constraint of the IRS reflection coefficient is always satisfied, P1 is converted into an unconstrained form, and the gradient descent algorithm is used to solve it. 7. The spatial multiplexing method based on the IRS-assisted terahertz MIMO communication system according to claim 1, characterized in that: in the step 6, based on mathematical derivation, a closed-form solution of the hybrid precoding matrix/combination matrix is obtained, and the cascaded channel is first decomposed by SVD

Where U is a unitary matrix of N _{r_tot} ×Q, Σ is a Q×Q diagonal matrix whose diagonal elements are the singular values of the concatenated channels, and V is a unitary matrix of N _{t_tot} ×Q.

Q is the rank of the cascade channel matrix H. By further analyzing the structure of the cascade channel matrix and simplifying the optimization problem of P2, the cascade channel matrix can be rewritten as follows:

in,

Combined with the SVD decomposition of H, the closed-form solution of the hybrid precoding matrix at the transmitter is obtained:

in,

represents the first N _s columns of the right singular matrix,

The normalized water injection power allocation matrix is:

represents the power allocated to the i-th data stream, and i＝1,2,L,N _s , ε is the water injection height,

Rewrite the cascaded channel matrix as:

in,

Combined with the SVD decomposition of H, the closed-form solution of the mixed combination code matrix at the receiving end is obtained

in,

Represents the first _Ns columns of the left singular matrix.

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