Beam Design Method for Maximizing User Sum Rate

Abstract

The present invention discloses a beam design method for maximizing the sum rate of users, which mainly solves the problems of low sum rate gain of a single intelligent reflecting surface and large number of elements of a double intelligent reflecting surface. The implementation scheme is as follows: The base station obtains an equivalent channel model through channel estimation, and performs Lagrangian and quadratic transformations on its maximum sum rate to obtain a polynomial representation of the maximum sum rate; designs a transmit beamforming matrix through weighted minimum mean square error, and respectively fixes the phase shift coefficients of the two intelligent reflecting surfaces and relaxes the non-convex constraints to obtain a semi-definite programming form of the maximum sum rate, and solves it to obtain the transmit beamforming matrix and the phase shift coefficients of the two intelligent reflecting surfaces for a single iteration; obtains the optimal values of the final transmit beamforming matrix and the phase shift coefficients of the two intelligent reflecting surfaces under the set iteration constraints. The present invention can obtain an obvious sum rate gain when the number of elements of the intelligent reflecting surface is the same and small, and can be used in cellular cell or local area network communication systems.

Description

Technical Field

[0001] The present invention belongs to the technical field of electronic circuits, and further relates to a beam design method for sum rate, which can be used in cellular cells or local area network communication systems. Background Art

[0002] It is expected that the capacity of communication networks will increase by 1000 times in the next decade, and ubiquitous wireless connectivity will become a reality. However, highly complex networks, high-cost hardware, and increasing energy consumption will be the key problems faced by future wireless communications. For example, in an ultra-dense network (UDN), a large number of base stations increase the hardware and maintenance costs and face severe network interference. The spectrum expands from sub-6 GHz to millimeter waves, and terahertz requires more complex signal processing and more expensive power-consuming hardware. It is imperative to research innovative, efficient, spectrum-saving, and resource-saving future wireless network solutions. The new technology of intelligent reflecting surface (IRS), which is an artificial electromagnetic surface structure with programmable electromagnetic characteristics, can independently control the electromagnetic characteristics of each element on the IRS through programming, thereby changing the phase or amplitude of the incident signal. By utilizing the ability of the IRS to reconstruct the wireless transmission environment, the signal transmission can be enhanced.

[0003] Currently, the IRS has been integrated into various communication systems, such as orthogonal frequency division multiplexing (OFDM) and non-orthogonal multiple access (NOMA), due to its unique characteristics of low cost, low power consumption, programmability, and easy deployment. Beamforming in an IRS-aided communication system is a technique for transmitting signals in a specific direction to enhance the signal, which directly affects the signal quality received by the user at the receiving end. Without beamforming, the base station will transmit signals in all directions, which will weaken the signal power received by the receiving user and thus reduce the performance of the system.

[0004] Huayan Guo et al. first proposed a beam design method for weighted sum rate maximization in a single IRS system in "Weighted Sum-Rate Maximization for Reconfigurable Intelligent Surface Aided Wireless Networks". First, the stationary solution of the joint design problem is obtained by using fractional programming, and then the stochastic successive convex approximation technique is adopted to achieve the beam design for weighted rate maximization in a single IRS system. Since this method only considers the scenario of a single IRS-aided communication, the system model established by it and the corresponding beam design method can only increase the received signal power at the receiving end by at most 2 dB when the number of IRS elements doubles.

[0005] For communication systems with relatively high requirements for communication capacity, due to the limitations of the IRS in terms of area and volume, the power gain provided by a single IRS in this method cannot meet the requirements of large-capacity communication systems.

[0006] Yitao Han et al. proposed a joint passive beamforming design method for two IRSs in "Cooperative Double-IRS Aided Communication: Beamforming Design and Power Scaling". Based on the geometric relationship between the two IRSs, it simulates the line-of-sight (LoS) propagation channel and obtains the passive beam matrix of the system through the conjugate zero-forcing method, thereby improving the signal-to-noise ratio (SNR) received by the user. However, the beam design algorithm proposed in this method can only demonstrate performance advantages when the number of IRS elements is relatively large. When the number of IRS elements is small, the performance gain provided by the two IRSs is lower than that of a single IRS. Therefore, this method can only demonstrate the performance advantage of double IRS over single IRS when the area and volume of the IRS are large. Summary of the Invention

[0007] The purpose of the present invention is to address the deficiencies of the above-mentioned existing technologies and propose a sum-rate beam design method applicable to users in a double-IRS system to achieve a higher power gain than a single IRS under the condition of a small number of IRS elements.

[0008] The technical idea of the present invention is as follows: For the sum-rate optimization problem of maximizing the user system with double IRSs, through Lagrangian dual transformation and quadratic transformation, this optimization problem is converted from a logarithmic summation problem into a polynomial summation problem; for the converted polynomial sum optimization problem, the beam design at the transmitter is carried out through the traditional weighted minimum mean square error (WMMSE); for the converted optimization model, the multi-variable problem is decoupled into a single-variable problem through an alternating optimization algorithm, and the original optimization problem is relaxed into a convex problem by relaxing non-convex constraints. At the same time, the problem is transformed into a classical semi-definite optimization problem through matrix Schur complement; the IRS phase shift coefficients are solved through the optimization tool of the computer, and by alternately optimizing the three variables in this optimization problem in each iteration process, the optimal values of the beam and phase shift coefficients are obtained under the specified number of iterations and iteration difference.

[0009] According to the above idea, the implementation steps of the present invention are as follows:

[0010] (1) Construct a baseband equivalent channel model and perform Lagrangian dual transformation and quadratic transformation on the maximum sum rate under this model:

[0011] 1a) The transmitting base station obtains the channel state information CSI through channel estimation, and obtains the initial phase shift coefficients Φ1 of the two IRSs through complex normal distribution (0) and Φ2 (0) , and obtains the baseband equivalent channel model through the channel state information CSI and the phase shift coefficients of the two IRSs

[0012] 1b) Substitute the baseband equivalent channel model into the signal-to-interference-plus-noise ratio SINR formula to obtain the signal-to-interference ratio γ of the user k , and obtain the user rate R according to γ k through the Shannon formula, and sum the user rates to obtain the expression form of the maximum sum rate;

[0013] 1c) Perform Lagrangian dual transformation on the expression form of the maximum sum rate by introducing the auxiliary variable α to convert the logarithmic sum into a fractional sum; then perform a quadratic transformation QT by introducing the auxiliary variable β to convert the fractional sum into a polynomial sum;

[0014] (2) Fix the phase shift coefficients of the two intelligent reflecting surfaces IRS, and obtain the transmit beamforming matrix W through the minimum mean square error beam design method;

[0015] (3) Convert the maximum sum rate including the phase shift coefficients Φ1 and Φ2 of the two intelligent reflecting surfaces IRS and the beamforming matrix W into the semidefinite programming SDP form:

[0016] 3a) Based on the beamforming matrix W, fix the phase shift coefficients Φ1 and Φ2 of the two intelligent reflecting surfaces IRS respectively, and by relaxing the non-convex constraint ||Φ i || = I to ||Φ i || ≤ I, convert the maximum sum rate with non-convex constraints into the maximum sum rate with convex constraints;

[0017] 3b) For the maximum sum rate with convex constraints, introduce the dual variable μ and perform Lagrangian dual decomposition to obtain the closed-form solution of the phase shift angle θ * of each element on the two IRSs;

[0018] 3c) Substitute θ * into the maximum sum rate form with convex constraints, and at the same time use the matrix Schur complement to convert this form into the classical semidefinite programming SDP form;

[0019] (4) Optimize the above semidefinite programming SDP form through the optimization tool CVX toolbox of the computer to obtain the optimal solutions of the phase shift coefficient Φ1 of the first intelligent reflecting surface IRS1, the phase shift coefficient Φ2 of the second intelligent reflecting surface IRS2, and the beamforming matrix W in a single iteration process;

[0020] (5) Let the maximum number of iterations be imax And the maximum iteration difference ε, to determine whether the termination iteration condition is satisfied:

[0021] When the iteration number i exceeds the set iteration number i max , or during two adjacent iteration processes, the difference between the optimal values of the objective function composed of Φ1, Φ2, and W is less than the maximum iteration difference ε, then the iteration ends, and the final optimized result W is obtained * , Φ1 * and Φ2 * ;

[0022] Otherwise, let the iteration number i = i + 1, and return to step (2).

[0023] The present invention has the following advantages compared with the prior art:

[0024] First, for the sum-rate maximization optimization problem of users under the double-intelligent reflecting surface IRS model, the present invention uses WMMSE to calculate the transmit beam matrix, and then fixes the phase shift coefficients of the two intelligent reflecting surfaces IRS respectively, and iteratively optimizes the transmit beam matrix and the phase shift coefficients of the two intelligent reflecting surfaces IRS. Compared with a single intelligent reflecting surface IRS-assisted communication system, under the condition of the same number of intelligent reflecting surface IRS elements, the double-intelligent reflecting surface IRS beam designed by the present invention can provide a higher system gain.

[0025] Second, for the sum-rate maximization optimization problem of users under the double-intelligent reflecting surface IRS model, the present invention uses Lagrangian dual transformation and quadratic transformation to convert the sum-rate form from logarithmic sum form to polynomial sum form. Compared with a single intelligent reflecting surface IRS-assisted communication system, this method has lower complexity. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 is the implementation flowchart of the present invention;

[0027] Figure 2 is the existing double-intelligent reflecting surface IRS-assisted communication system model diagram used in the present invention;

[0028] Figure 3 is the sum-rate comparison diagram between the double-intelligent reflecting surface IRS and the existing single-intelligent reflecting surface IRS of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0029] The following further describes the embodiments and effects of the present invention with reference to the drawings.

[0030] Refer to Figure 2, the dual-intelligent reflecting surface (IRS)-assisted communication system used in the present invention includes an access point (AP), L user receivers, a first intelligent reflecting surface IRS1, and a second intelligent reflecting surface IRS2. The second intelligent reflecting surface IRS2 is deployed near the access point AP, and the first intelligent reflecting surface IRS1 is deployed near the users. The direct link between the AP and the users is blocked, and signals can only be transmitted through the two distributed intelligent reflecting surfaces IRS. The system not only has links from each intelligent reflecting surface IRS to the users, but also has a reflection link between the two intelligent reflecting surfaces IRS1 and IRS2. The two intelligent reflecting surfaces IRS1 and IRS2 have the same structure. The access point AP adopts a uniform linear array (ULA), and the reflection coefficients of the units of the two intelligent reflecting surfaces are controlled by an intelligent controller to achieve the passive beam design of the system, thereby reconstructing the propagation environment between the users and the access point AP.

[0031] Referring to Figure 1 , based on the above system, the beam design method of this example is implemented as follows:

[0032] Step 1: By obtaining the channel state information and the initial phase shift coefficients of the two intelligent reflecting surfaces IRS, a baseband equivalent channel model is constructed.

[0033] 1.1) The base station obtains the channel state information CSI through channel estimation. The initial phase shift coefficients of the two intelligent reflecting surfaces IRS are obtained through complex normal distribution, that is, the initial phase shift coefficient Φ1 of the first intelligent reflecting surface IRS1 (0) and the initial phase shift coefficient Φ2 of the second intelligent reflecting surface IRS2 (0) . Through the channel state information CSI and the phase shift coefficients of the two intelligent reflecting surfaces, the baseband equivalent channel model is obtained

[0034]

[0035] where represents the channel from the first intelligent reflecting surface IRS1 to user k, represents the channel from the second intelligent reflecting surface IRS2 to user k, represents the channel from the second intelligent reflecting surface IRS2 to the first intelligent reflecting surface IRS1, represents the channel from the access point AP to the first intelligent reflecting surface IRS1, represents the channel from the access point AP to the second intelligent reflecting surface IRS2, M1 represents the maximum number of elements of the first intelligent reflecting surface IRS1, M2 represents the maximum number of elements of the second intelligent reflecting surface IRS2, N represents the number of antennas of the access point AP, and [·] H represents the conjugate transpose of the matrix.

[0036] 1.2) According to the baseband equivalent channel model obtain the signal y received by user k k :

[0037]

[0038] where is the beamforming vector of user k, is the beamforming vector of interfering user j, u k is a complex Gaussian noise that follows , is the data symbol received by user k from the access point AP side, t k ω k s k is the signal received by user k, is the interference from other users received by user k.

[0039] Step 2, construct the maximum sum-rate expression through the baseband equivalent channel model.

[0040] 2.1) According to the baseband equivalent channel model express the signal-to-interference-plus-noise ratio γ k of user k in the system as:[[]]

[0041]

[0042] where σ0 2 represents the noise power that follows the complex symmetric complex Gaussian (CSCG) distribution;

[0043] 2.2) According to the signal-to-interference-plus-noise ratio γ k of user k, obtain the user rate R through the Shannon formula:[[]]

[0044] R = log2(1 + γ k )

[0045] 2.3) According to the user rate R and the weight w k of user k, obtain the expression of the system sum-rate R sum :[[]]

[0046]

[0047] 2.4) According to the characteristics that the intelligent reflecting surface (IRS) only changes the phase and does not change the amplitude, and the characteristics of the transmission power limitation, use the system sum-rate R sum to obtain the expression P1 of the original maximum sum-rate form:[[]]

[0048] P1:

[0049]

[0050]

[0051] W = [ω1, ω2, …, ω k , …, ω L is the transmit beamforming matrix, L is the total number of users in the system, and P T represents the transmit power constraint of the AP; Φ1 is the phase shift coefficient of the first intelligent reflecting surface IRS1, Φ1 = diag(θ1), where θ1 represents the phase shift angle of each reflecting element on the first intelligent reflecting surface IRS1, and M1 represents the maximum number of elements on the first intelligent reflecting surface IRS1; Φ2 = diag(θ2) is the phase shift coefficient of the second intelligent reflecting surface IRS2, where θ2 represents the phase shift coefficient of each reflecting element on the second intelligent reflecting surface IRS2, and M2 represents the maximum number of elements on the second intelligent reflecting surface IRS2.

[0052] Step 3: Perform Lagrangian transformation and quadratic transformation on the original maximum sum rate form to obtain the transformed maximum sum rate form P1”.

[0053] 3.1) Introduce the auxiliary variable α = [α1, α2, …, α k , …, α L T , and perform Lagrangian dual transformation on P1 to obtain the Lagrangian-transformed maximum sum rate form P1':

[0054] P1':

[0055]

[0056]

[0057] where represents the sum rate, L represents the total number of users in the system, k represents the current user index, and [·] T represents the transpose of the matrix;

[0058] 3.2) Take the partial derivative of the sum rate with respect to α k and set the partial derivative to 0, i.e., to obtain Then substitute into the formula, and the sum rate is simplified to

[0059]

[0060] where α k ​Auxiliary variables introduced for the Lagrangian transform;

[0061] 3.3) For the simplified sum rate Introduce an auxiliary variable β for quadratic transformation to obtain the sum rate after quadratic transformation

[0062]

[0063] where the auxiliary variable β = [β1, β2, …, β k , …, β L T , is the auxiliary variable for the k-th user, L is the total number of users in the system, and α k is the auxiliary variable introduced for the Lagrangian transform;

[0064] 3.4) Substitute f y1 (W, Φ1, Φ2, β) into the original maximum sum rate form P1 to obtain the maximum sum rate form P1″ after Lagrangian transform and quadratic transformation:

[0065] P1″:

[0066]

[0067]

[0068] Step 4, fix the phase shift coefficients Φ1 and Φ2 of the two intelligent reflecting surfaces to obtain the transmit beam ω k in closed form.

[0069] 4.1) Fix the phase shift coefficient Φ1 of the first intelligent reflecting surface IRS1 and the phase shift coefficient Φ2 of the second intelligent reflecting surface IRS2 as constants, so that the maximum sum rate form P1″ after quadratic transformation degenerates into a sum rate form P2 that depends only on the transmit beamforming matrix W, and its form is as follows:

[0070] P2:

[0071]

[0072] 4.2) By using the Lagrange multiplier method, introduce the Lagrange dual variable λ to obtain the closed form of ω k as:

[0073]

[0074] where is the equivalent baseband model of the interfering user i, and I N represents the identity matrix of size N×N.​

[0075] Step 5, solve the transmit beamforming matrix W.

[0076] 5.1) For the transmit beam ω of user k k Use the bisection method to obtain the optimal solution of the Lagrangian dual variable λ as λ * , and substitute λ * back into the closed-form solution of the transmit beam ω of user k k to obtain the optimal value of the transmit beam ω of user k in a single optimization; k

[0077] 5.2) Concatenate the transmit beams ω1~ω of L users L to obtain the transmit beamforming matrix W.

[0078] Step 6, fix the phase shift coefficient Φ1 of the first intelligent reflecting surface IRS1, and obtain the quadratic constraint and quadratic programming QCQP form P3” of the phase shift coefficient vector θ2 of the reflection units of the second intelligent reflecting surface IRS2 under convex constraints.

[0079] 6.1) Fix the phase shift coefficient Φ1 of the second intelligent reflecting surface IRS1, and rewrite the maximum sum rate form P1” after the quadratic transformation as the maximum sum rate form P3 of the phase shift coefficient vector θ2 of the reflection units of the second intelligent reflecting surface IRS2:

[0080] P3:

[0081]

[0082] where, (·) * denotes the conjugate of a complex number.

[0083] 6.2) Perform mathematical transformation on the baseband equivalent model obtained in step 1 to obtain the following form of the baseband equivalent channel t k ' of the second intelligent reflecting surface IRS2:

[0084]

[0085] where r represents the index of the current intelligent reflecting surface IRS, diag(·) denotes the matrix diagonalization operation on a vector, and M r is the number of elements of the r-th intelligent reflecting surface IRS, denotes the value after the last update during the iteration;

[0086] 6.3) The baseband equivalent channel t of the second intelligent reflecting surface IRS2 kSubstitute it into the maximum sum rate form \(P3\) of the phase shift coefficient vector \(\theta_2\) of the second intelligent reflecting surface IRS2, and obtain the maximum sum rate function \(f\) of the phase shift coefficient vector \(\theta_2\) of the second intelligent reflecting surface IRS2 b1 (\(\theta_2\)):

[0087]

[0088] where

[0089]

[0090] 6.4) For the maximum sum rate function of the phase shift coefficient vector \(\theta_2\) of the second intelligent reflecting surface IRS2 Perform variable substitution, delete irrelevant constant terms, and rewrite the maximum sum rate form \(P3\) of the phase shift coefficient vector \(\theta_2\) of the second intelligent reflecting surface IRS2 into the quadratic constraint quadratic programming QCQP form \(P3'\) of the phase shift coefficient vector \(\theta_2\) of the second intelligent reflecting surface IRS2:

[0091] \(P3'\):

[0092]

[0093] where

[0094]

[0095]

[0096] \(q_2\) is the first-order coefficient in the maximum sum rate form with variable \(\theta_2\), \(F_2\) is the second-order coefficient in the maximum sum rate form with variable \(\theta_2\); \(k\) is the current user index, \(r\) represents the index of the current IRS, \(\theta_1\) represents the phase shift coefficient vector of the first intelligent reflecting surface IRS1 reflection unit, \(\omega\) k is the beamforming vector of user \(k\), \(\omega\) j is the beamforming vector of interfering user \(j\), \(\alpha\) is the auxiliary variable introduced by the Lagrangian transform, \(\beta\) is the auxiliary variable introduced by the quadratic transform, represents the quadratic form of the system sum rate with the phase shift coefficient vector \(\theta_2\) of the second intelligent reflecting surface IRS2 as the variable;

[0097] 6.5) Relax the non-convex constraints in the quadratic constraint quadratic programming form \(P3'\) of the single variable \(\theta_2\), and convert \(P3'\) into the QCQP form \(P3''\) of the phase shift coefficient vector \(\theta_2\) of the second intelligent reflecting surface IRS2 under convex constraints:

[0098] \(P3''\):

[0099]

[0100] where e m is the initial vector with 1 at the m-th position, m represents the element index on the intelligent reflecting surface IRS, r represents the index of the current intelligent reflecting surface IRS, and M r is the number of elements of the r-th intelligent reflecting surface IRS.

[0101] Step 7: Fix the phase coefficient Φ2 of the second intelligent reflecting surface IRS2 to obtain the quadratic constraint quadratic programming QCQP form P4” of the phase shift coefficient vector θ1 of the reflection units of the first intelligent reflecting surface IRS1 under convex constraints.

[0102] 7.1) Fix the phase shift coefficient Φ2 of the second intelligent reflecting surface IRS2, and rewrite the maximum sum rate form P1” after quadratic transformation as the maximum sum rate form P4 of the phase shift coefficient vector θ1 of the reflection units of the first intelligent reflecting surface IRS1:

[0103] P4:

[0104]

[0105] where f c1 (θ1) is the maximum sum rate function of the phase shift coefficient vector θ1 of the reflection units of the first intelligent reflecting surface IRS1.

[0106] 7.2) Perform mathematical transformation on the baseband equivalent model obtained in Step 1 to obtain the baseband equivalent channel t k ” in the following form:

[0107]

[0108] where and r represents the index of the current intelligent reflecting surface IRS, and M r is the number of elements of the r-th intelligent reflecting surface IRS;

[0109] 7.3) Substitute the baseband equivalent channel t k ' of the first intelligent reflecting surface IRS1 into the maximum sum rate form P4 of the phase shift coefficient vector θ1 of the reflection units of the first intelligent reflecting surface IRS1. At the same time, perform variable substitution and delete irrelevant constant terms, and rewrite the maximum sum rate form P4 of the phase shift coefficient vector θ1 of the reflection units of the first intelligent reflecting surface IRS1 as the quadratic constraint quadratic programming QCQP form P4' of the phase shift coefficient vector θ1 of the reflection units of the first intelligent reflecting surface IRS1:

[0110] P4':

[0111]

[0112] Among them

[0113]

[0114]

[0115] q1 is the first-order coefficient in the maximum sum rate form with the variable θ1, and F1 is the second-order coefficient in the maximum sum rate form with the variable θ1 represents the quadratic form of the system sum rate with the variable being the phase shift coefficient vector θ1 of the first intelligent reflecting surface IRS1

[0116] 7.4) Relax the non-convex constraints in the quadratic constrained quadratic programming form P4' of the phase shift coefficient vector θ1 of the first intelligent reflecting surface IRS1, and convert P4' into the QCQP form P4” of the phase shift coefficient vector θ1 of the first intelligent reflecting surface IRS1 under convex constraints

[0117] P4”:

[0118]

[0119] where e m is the initial vector with 1 in the m-th position, and m represents the element index on the intelligent reflecting surface IRS

[0120] Step 8, obtain the closed-form solutions of the phase shift coefficient vectors θ * of the reflecting units on the two intelligent reflecting surfaces IRS through Lagrangian dual decomposition

[0121] 8.1) Perform Lagrangian dual decomposition on the QCQP form P3″ of the phase shift coefficient vector θ2 of the second intelligent reflecting surface IRS2 under convex constraints to obtain the QCQP form P3”' of the phase shift coefficient vector θ2 of the second intelligent reflecting surface IRS2 after Lagrangian dual decomposition

[0122] P3”':

[0123]

[0124] where represents the Lagrangian function of the QCQP form of the phase shift coefficient vector θ2 of the second intelligent reflecting surface IRS2 represents the dual variable of the second intelligent reflecting surface, and μ 2,m is the dual variable of the m-th element on the second intelligent reflecting surface, and m represents the element index on the intelligent reflecting surface IRS Denote the quadratic form of the system sum rate with the phase shift coefficient vector θ2 of the reflection elements of the second intelligent reflecting surface IRS2 under convex constraints as the variable.

[0125] 8.2) Perform Lagrangian dual decomposition on the QCQP form P4” of the phase shift coefficient vector θ1 of the reflection elements of the first intelligent reflecting surface IRS1 under convex constraints. The QCQP form P4”' of θ1 after Lagrangian dual decomposition:

[0126] P4”':

[0127]

[0128] where Denote the Lagrangian function of the QCQP form of the phase shift coefficient vector θ1 of the reflection elements of the first intelligent reflecting surface IRS1, where Denote the dual variable of the first intelligent reflecting surface, μ 1,m is the dual variable of the m-th element on the first intelligent reflecting surface, and m represents the element index on the intelligent reflecting surface IRS. Denote the quadratic form of the system sum rate with the phase shift coefficient vector θ1 of the reflection elements of the first intelligent reflecting surface IRS1 as the variable;

[0129] 8.3) Take the partial derivative of the Lagrangian function Λ(θ2, μ2) of the QCQP form of the phase shift coefficient vector θ2 of the reflection elements of the second intelligent reflecting surface IRS2 with respect to μ2 to obtain the optimal closed-form solution θ2 of θ2 * is:

[0130]

[0131] 8.4) Take the partial derivative of the Lagrangian function Λ(θ1, μ1) of the QCQP form of the phase shift coefficient vector θ1 of the reflection elements of the first intelligent reflecting surface IRS1 with respect to μ1 to obtain the optimal closed-form solution of θ1 as

[0132]

[0133] Step 9, through the matrix Schur complement, obtain the semi-definite programming SDP form of the maximum sum rate of the two intelligent reflecting surfaces IRS.

[0134] 9.1) Substitute the above into the QCQP form P3”' of the phase shift coefficient vector θ2 of the reflection elements of the second intelligent reflecting surface IRS2 after Lagrangian dual decomposition, and substitute the above into the QCQP form P4”' of the phase shift coefficient vector θ1 of the reflection elements of the first intelligent reflecting surface IRS1 after Lagrangian dual decomposition to obtain the sum rate τ under the semi-definite programming of the r-th intelligent reflecting surface r :

[0135] τ r = q r H (F r + diag(μ r ))q + tr(diag(μ r ))

[0136] where r represents the index of the current IRS;

[0137] 9.2) Sum rate τ under semidefinite programming r Use the matrix Schur complement form for transformation and substitute it into the QCQP form P3''' of the phase shift coefficient vector θ2 of the second intelligent reflecting surface IRS2 after Lagrangian dual decomposition to obtain the semidefinite programming SDP form P3 for maximizing the sum rate of the second intelligent reflecting surface IRS2 final It is expressed as:

[0138] P3 final :

[0139]

[0140] where τ2 is the sum rate under semidefinite programming of the second intelligent reflecting surface;

[0141] 9.3) Sum rate τ under semidefinite programming r Use the matrix Schur complement form for transformation and substitute it into the QCQP form P4''' of the phase shift coefficient vector θ1 of the first intelligent reflecting surface IRS1 after Lagrangian dual decomposition to obtain the semidefinite programming SDP form P4 for maximizing the sum rate of the first intelligent reflecting surface IRS1 final It is expressed as:

[0142] P4 final :

[0143]

[0144] where τ1 is the sum rate under semidefinite programming of the first intelligent reflecting surface.

[0145] Step 10, use the computer optimization tool CVX toolbox to obtain the optimal solution Φ2 of the phase shift coefficients of the second intelligent reflecting surface IRS2 and the optimal solution Φ1 of the phase shift coefficients of the first intelligent reflecting surface IRS1.

[0146] 10.1) Use the computer optimization tool CVX toolbox for the semidefinite programming SDP form P3 for maximizing the sum rate of the second intelligent reflecting surface IRS2 finalOptimize to obtain the optimal solution Φ2 of the phase shift coefficient of the second intelligent reflecting surface IRS2 in a single iteration process;

[0147] 10.2) Use the optimization tool CVX toolbox of the computer to optimize the semi - definite programming SDP form P4 of maximizing the sum rate of the second intelligent reflecting surface IRS2 final Optimize to obtain the optimal solution Φ1 of the phase shift coefficient of the first intelligent reflecting surface IRS1 in a single iteration process.

[0148] Step 11, judge the iteration termination condition to obtain the final optimization result W * , Φ1 * and Φ2 * .

[0149] 11.1) Set the maximum number of iterations i max and the maximum iteration difference ε;

[0150] 11.2) Substitute the optimal solution Φ1 of the phase shift coefficient of the first intelligent reflecting surface IRS1, the optimal solution Φ2 of the phase shift coefficient of the second intelligent reflecting surface IRS2, and the transmit beamforming matrix W into the objective function to obtain the sum rate in this iteration process;

[0151] 11.3) When the number of iterations i exceeds the set number of iterations i max , or in two adjacent iteration processes, the difference between the sum rates of the objective function constituted by Φ1, Φ2, and W is less than the maximum iteration difference ε, then the iteration ends, and the final optimization result W * , Φ1 * and Φ2 * are obtained, and the beam design for maximizing the user sum rate is completed;

[0152] 11.4) Otherwise, let the number of iterations i = i + 1, and return to step 4.

[0153] The effects of the present invention can be further illustrated by the following simulation results

[0154] I. Simulation conditions

[0155] Set the coordinate position of the access point AP as (0m, 0m), the coordinate position of the center of the user cluster as (40m, 1m), the radius r of the user cluster = 0.5m, the coordinate position of the second intelligent reflecting surface IRS2 as (1m, 1m), and the coordinate position of the first intelligent reflecting surface IRS1 as (39m, 1m);

[0156] Set the number of antennas N of the access point AP as 32, the total transmit power as P T = 0dBm, and the noise power of the user receiver as σ0 = - 85dBm;

[0157] Set the weights w of all users k to be equal.

[0158] II. Simulation content

[0159] Under the above simulation conditions, the sum rate of the maximum system achieved by designing beams with the present invention and the existing method is respectively simulated, and the results are as Figure 3 ,

[0160] There is Figure 3 It can be seen that, under the condition of the same number of intelligent reflecting surface (IRS) elements, the double IRS of the present invention has an obvious sum rate gain compared with the existing single IRS, and as the number of IRS elements increases, the sum rate gain of the double IRS over the single IRS becomes more obvious.

[0161] The above description is only a specific example of the present invention and does not constitute any limitation to the present invention. Obviously, for professionals in the field, after understanding the content and principle of the present invention, various modifications and changes in form and details may be made without departing from the principle and structure of the present invention. However, these corrections and changes based on the idea of the present invention are still within the scope of protection of the claims of the present invention.

Claims

1. A beam design method for maximizing the sum rate of users, characterized in that It includes the following steps: (1) Construct a baseband equivalent channel model, and perform Lagrangian dual transformation and quadratic transformation on the maximum sum rate under this model: 1a) The transmitting base station obtains the channel state information CSI through channel estimation, and obtains the initial phase shift coefficients Φ1 of two intelligent reflecting surfaces IRS through complex normal distribution (0) and Φ2 (0) , and obtains the baseband equivalent channel model through the channel state information CSI and the phase shift coefficients of the two IRSs 1b) Substitute the baseband equivalent channel model into the signal-to-interference-plus-noise ratio (SINR) formula to obtain the signal-to-interference ratio γ of the user k , according to γ k Use the Shannon formula to obtain the user rate R, and sum the user rates to obtain the expression form of the maximum sum rate; 1c) Perform Lagrangian dual transformation on the representation form of the maximum sum rate by introducing an auxiliary variable α to convert the logarithmic sum into a fractional sum; then perform quadratic transformation QT by introducing an auxiliary variable β to convert the fractional sum into a polynomial sum, which is expressed as follows: where β = [β1, β2, …, β k , …, β L T is the auxiliary variable introduced by the second transformation in step 1c), L is the total number of users in the system, and γ k represents the signal-to-interference-plus-noise ratio of user k, α k is the auxiliary variable introduced by the Lagrangian transformation, is the equivalent baseband model of user k, ω k is the beamforming vector of user k, ω j is the beamforming vector of interfering user j, σ0 2 is the noise power, [·] T represents the transpose of the matrix;​ (2) Fix the phase shift coefficients of two intelligent reflecting surfaces (IRSs), and obtain the transmit beamforming matrix W through the minimum mean square error beam design method, as follows: 7a) Let the initial phase shift coefficient of the first intelligent reflecting surface IRS1 be Φ1 (0) , and the initial phase shift coefficient of the second intelligent reflecting surface IRS2 be Φ2 (0) . Fix them as constants, introduce the Lagrange dual variable λ, and obtain the closed-form solution of the transmit beam ω of user k containing λ through the Lagrange multiplier method k ; wherein α k is an auxiliary variable introduced by Lagrangian transformation in step 1c), and β k is an auxiliary variable introduced by quadratic transformation in step 1c), is the equivalent baseband model of user k, is the equivalent baseband model of interfering user i, and I N represents an N×N identity matrix; 7b) Transmit beam ω for user k obtained in 7a) k The closed-form solution of, use the bisection method to obtain the optimal solution λ * , substitute λ * back into the closed-form solution of the transmit beam ω for user k k , cascade the transmit beams of L users to obtain the transmit beamforming matrix W; (3) Convert the maximum sum rate including the phase shift coefficients Φ1 and Φ2 of two intelligent reflecting surfaces (IRSs) and the beamforming matrix W into the form of semidefinite programming (SDP): 3a) Based on the beamforming matrix W, fix the phase shift coefficients Φ1 and Φ2 of the two intelligent reflecting surfaces (IRSs) respectively. By relaxing the non-convex constraint ||Φ i || = I to ||Φ i || ≤ I, the maximum sum rate with non-convex constraints is transformed into the maximum sum rate with convex constraints, which is expressed as follows: where q1 is the first-order coefficient in the maximum sum rate form with variable θ1, q2 is the first-order coefficient in the maximum sum rate form with variable θ2, F1 is the second-order coefficient in the maximum sum rate form with variable θ1, F2 is the second-order coefficient in the maximum sum rate form with variable θ2; e m is the initial vector with 1 at the m-th bit, $\text{diag}(\cdot)$ represents the matrix diagonalization operation on a vector, $k$ is the current user index, $r$ is the index of the current intelligent reflecting surface (IRS), and $M$ r is the number of elements of the $r$-th IRS, $\boldsymbol{\theta}_2$ represents the reflection element phase shift coefficient vector of the second IRS ($\text{IRS}_2$), $\boldsymbol{\theta}_1$ represents the reflection element phase shift coefficient vector of the first IRS ($\text{IRS}_1$), $\alpha$ is the auxiliary variable introduced by the Lagrangian transformation, and $\beta$ is the auxiliary variable introduced by the quadratic transformation. denotes the value after the last update during the iterative process; 3b) For the maximum sum rate with convex constraints, introduce the dual variable μ and perform Lagrangian dual decomposition to obtain the closed-form solution of the phase shift coefficient vector θ of the reflecting elements on the two intelligent reflecting surfaces (IRSs). * ​ 3c) Substitute θ * into the maximum sum rate form with convex constraints, and at the same time use the matrix Schur complement to transform this form into the classical semi-definite programming SDP form; (4) Optimize the above semidefinite programming (SDP) form through the optimization tool CVX toolbox of the computer to obtain the phase shift coefficient Φ1 of the first intelligent reflecting surface IRS1 and the phase shift coefficient Φ2 of the second intelligent reflecting surface IRS2 in a single iteration process; (5) Set the maximum number of iterations \(i\). max And the maximum iteration difference \(\varepsilon\), and determine whether the termination iteration condition is satisfied: When the iteration number i exceeds the set iteration number i max , or during two adjacent iteration processes, the difference between the sum rates composed of Φ1, Φ2, and W is less than the maximum iteration difference ε, then the iteration ends and the final optimized result W is obtained * , and Otherwise, let the iteration number i = i + 1, and return to step (2).

2. The method according to claim 1, wherein The baseband equivalent channel model obtained in step 1a) is represented as follows: Among them represents the channel from the first intelligent reflecting surface IRS1 to user k represents the channel from the second intelligent reflecting surface IRS2 to user k represents the channel from the second intelligent reflecting surface IRS2 to the first intelligent reflecting surface IRS1 represents the channel from the access point AP to the first intelligent reflecting surface IRS1 represents the channel from the access point AP to the second intelligent reflecting surface IRS2; Φ1 is the phase shift coefficient of the first intelligent reflecting surface IRS1, and Φ2 is the phase shift coefficient of the second intelligent reflecting surface IRS2; [·] H represents the conjugate transpose of a matrix 3. The method according to claim 1, characterized in that The signal-to-interference-plus-noise ratio γ obtained in step 1b) k and the user rate R are respectively expressed as follows: R = log2(1 + γ k ) where is the beamforming vector of user k, is the beamforming vector of interfering user j, is the equivalent baseband model of user k obtained in step 1a), and σ0 2 represents the noise power that follows a complex symmetric complex Gaussian (CSCG) distribution.

4. The method according to claim 1, characterized in that, The expression form of the maximum sum rate obtained in step 1b) is expressed as follows: where ω k is the beamforming vector of user k, W = [ω1, ω2, …, ω k , …, ω L is the transmit beamforming matrix, L is the total number of users in the system, w k is the weight of user k, γ k represents the signal-to-interference-plus-noise ratio (SINR) of user k, P T represents the transmit power constraint of the access point (AP); Φ1 is the phase shift coefficient of IRS1, Φ1 = diag(θ1), where θ1 represents the phase shift angle of each reflecting element on IRS1, and M1 represents the maximum number of elements on IRS1; Φ2 = diag(θ2) is the phase shift coefficient of IRS2, where θ2 represents the phase shift coefficient of each reflecting element on IRS2, and M2 represents the maximum number of elements on IRS2.

5. The method according to claim 1, wherein In step 1c), through Lagrangian transformation, the logarithmic sum is converted into a fractional sum, which is expressed as follows: where α k is an auxiliary variable introduced by Lagrangian transformation in step 1c), is the equivalent baseband model of user k, ω k is the beamforming vector of user k, ω j is the beamforming vector of interfering user j, w k is the weight, σ0 2 is the noise power, γ k represents the signal-to-interference ratio of user k.

6. The method according to claim 1, wherein The closed-form solutions of the reflection element phase shift coefficient vectors θ on the two intelligent reflecting surfaces (IRSs) obtained in step 3b) are expressed as follows: * ​ where θ1 * is the phase shift coefficient vector of the reflection elements of the first intelligent reflecting surface IRS1, and θ2 m is the initial vector with 1 at the m-th position, where m represents the element index on the intelligent reflecting surface IRS, μ 2,m is the dual variable of the m-th element on the second intelligent reflecting surface, and μ 1,m is the dual variable of the m-th element on the first intelligent reflecting surface. M2 represents the maximum number of elements on IRS2, M1 is the number of elements of IRS1, M2 is the number of elements of IRS2, q1 is the first-order coefficient in the maximum sum rate form with variable θ1, q2 is the first-order coefficient in the maximum sum rate form with variable θ2, F1 is the second-order coefficient in the maximum sum rate form with variable θ1, and F2 is the second-order coefficient in the maximum sum rate form with variable θ2.

7. The method according to claim 1, characterized in that, The classical semidefinite programming (SDP) form obtained in step 3c) is expressed as follows: τ r = q r H (F r + diag(μ r ))q r + tr(diag(μ r )) ; where τ r is the sum rate under the semidefinite programming of the r-th intelligent reflecting surface (IRS), denotes the dual variable introduced in the Lagrangian dual decomposition, r represents the index of the current IRS, M r represents the number of elements on the r-th IRS, m represents the element index on the IRS, q r is the first-order coefficient in the maximum sum rate form with variable θ r , F r is the second-order coefficient in the maximum sum rate form with variable θ r , and [·] T denotes the transpose of a matrix.

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