Beamforming method for active RIS-ISAC system to prevent eavesdropping and interference

Abstract

This invention discloses an active RIS-ISAC system beamforming method for preventing eavesdropping and interference. It establishes a system model including a dual-function radar-communication base station, an active RIS, legitimate users, illegal eavesdroppers, and jammers, and sets communication, sensing, and power consumption performance indicators. Then, it constructs an optimization problem that aims to minimize the total system power consumption while satisfying communication and sensing constraints, jointly optimizing the base station's transmit beamforming vector and the active RIS's phase shift matrix. The non-convex optimization problem is further decomposed into two sub-problems, and the optimal configurations of the base station and RIS are solved separately. Finally, an alternating optimization algorithm is used to obtain the optimal solution. The advantage is that, even in the presence of both illegal eavesdroppers and malicious adversaries, it achieves improved communication performance, suppression of eavesdropping and interference, and a reduction in total system power consumption by jointly designing the base station's transmit beamforming vector and the active RIS's phase shift matrix.

Description

Technical Field

[0001] This invention relates to an active RIS-ISAC (Reconfigurable Intelligent Surface Integrated Sensing and Communication) system, and more particularly to a beamforming method for an active RIS-ISAC system to prevent eavesdropping and interference. Under simultaneous eavesdropping and malicious interference attacks, the method aims to minimize the total system power consumption. It considers the imperfections of the Channel State Information (CSI) of illegal nodes (illegal eavesdroppers and enemy targets, i.e., jammers). Under constraints of the achievable communication rate of legitimate users, the eavesdropping rate of illegal eavesdroppers, and the beam gain generated in the direction from the active RIS (reconfigurable intelligent surface) to the enemy target, the method designs the base station's transmit beamforming vector and the active RIS's phase shift matrix to achieve the goal of the base station simultaneously performing secure communication and target sensing. Background Technology

[0002] The widespread adoption of wireless communication and radar systems has exacerbated frequency saturation issues, raising concerns about spectrum over-congestion and congestion. Next-generation wireless communication technologies require not only ultra-high-speed communication rates but also high-precision sensing services. To address this challenge, ISAC (Integrated Communication and Sensing) has emerged as a revolutionary paradigm. Its concept is gaining increasing attention from academia, industry, and standardization bodies, and has been formally included in the six major use cases of 6G by the ITU-R. ISAC technology shares spectrum, hardware platforms, and even waveform and signal processing between communication and sensing, thereby improving the system's integrated gain. ISAC systems can perform tasks such as target sensing and localization tracking while meeting traditional communication requirements.

[0003] Due to the broadcast nature of wireless channels and the spectrum-sharing nature of communication sensing, ISAC systems are vulnerable to attacks from malicious systems, such as interference from adversary targets (jammers), eavesdropping by unauthorized users, and deception. Existing research on physical layer security issues in ISAC systems largely focuses on eavesdroppers intercepting legitimate data, for example, by emitting artificial noise to counter this threat. However, in real-world electromagnetic environments, malicious jammers may also send interference signals to disrupt legitimate communication and sensing. Existing beamforming designs cannot address this more challenging scenario. Therefore, when both unauthorized eavesdroppers and jammers exist within an ISAC system, designing an integrated sensing beamforming method that simultaneously prevents eavesdropping and interference is crucial.

[0004] RIS (Reconfigurable Smart Surface), as a revolutionary low-cost technology, can help wireless communication systems and sensing systems achieve intelligent and reconfigurable propagation environments. It is considered a key enabling technology in ISAC systems, solving problems such as limited coverage and blocked links in ISAC systems.

[0005] Active RIS, as an extension of traditional passive RIS, can amplify signals while performing phase modulation, thereby enhancing signal reception strength and solving the signal attenuation problem caused by "multiplicative fading" in passive RIS. Furthermore, due to the potential unknowability of illicit nodes, the CSI (Channel State Information) associated with illegal eavesdroppers and jammers may not be fully available to legitimate systems. Existing physical layer security studies of active RIS-ISAC systems largely assume that the system can obtain perfect CSI, which may be limiting in practical applications. Therefore, a robust beamforming method should be designed to address the complexities of situations where CSI associated with illegal nodes (illegal eavesdroppers and jammers) cannot be perfectly obtained. Summary of the Invention

[0006] The technical problem to be solved by this invention is to provide an active RIS-ISAC system beamforming method for preventing eavesdropping and interference. In the case of the simultaneous presence of illegal eavesdroppers and malicious enemy targets, and considering the imperfect channel state information of illegal nodes, this method improves communication performance, suppresses eavesdropping and interference, and reduces the total power consumption of the system by jointly designing the base station's transmit beamforming vector and the phase shift matrix of the active RIS.

[0007] The technical solution adopted by this invention to solve the above-mentioned technical problems is: an active RIS-ISAC system beamforming method for preventing eavesdropping and interference, characterized by including the following steps:

[0008] Step 1: Establish an active RIS-ISAC system model. This model includes a base station with dual radar and communication functions, an active RIS, a legitimate user, an illegal eavesdropper, and an enemy target, i.e., a jammer. With the assistance of the active RIS, the base station communicates with the legitimate user via downlink. The illegal eavesdropper attempts to steal the signal sent by the base station to the legitimate user. The base station detects the enemy target, and the enemy target will actively send jamming signals to the legitimate user to interfere with the legitimate user and destroy the communication between the base station and the legitimate user.

[0009] Step 2: Establish the communication performance indicators, sensing performance indicators, and power consumption performance indicators of the active RIS-ISAC system model. The communication performance indicators include the reachable communication rate of legitimate users and the eavesdropping rate of illegal eavesdroppers. The sensing performance indicators include the beam gain generated by the active RIS in the direction to the enemy target. The power consumption performance indicators include the total power consumption of the system.

[0010] Step 3: With the goal of minimizing the total power consumption of the system, and constrained by the reachable communication rate of legitimate users, the eavesdropping rate of illegal eavesdroppers, and the beam gain generated in the direction of the active RIS to the enemy target, jointly optimize the base station's transmit beamforming vector and the phase shift matrix of the active RIS to construct an optimization problem;

[0011] Step 4: Decompose the optimization problem into subproblem 1 and subproblem 2. Subproblem 1 is used to optimize the transmit beamforming vector of the base station with a fixed phase shift matrix of the active RIS. Subproblem 2 is used to optimize the phase shift matrix of the active RIS with a fixed transmit beamforming vector of the base station. Then, subproblem 1 is transformed into convex problem 1 and subproblem 2 is transformed into convex problem 2.

[0012] Step 5: Use the alternating optimization algorithm to iteratively solve convex problem 1 and convex problem 2 to obtain the optimal solutions for the base station's transmit beamforming vector and the phase shift matrix of the active RIS.

[0013] In step 1, the base station is equipped with a uniform linear array with N antennas; the number of reflective elements of the active RIS is M; and both legitimate users and illegal eavesdroppers are equipped with a single antenna.

[0014] In step 2, the process of obtaining the reachable rate of a legitimate user is as follows:

[0015] Step 2.1a: Transfer the received signal y from the legitimate user u Represented as Wherein, the superscript "H" indicates the conjugate transpose operation, h R Let Φ represent the channel from the active RIS to the legitimate user, and let Φ represent the phase shift matrix of the active RIS. The `diag(·)` function is used to extract or construct the diagonal elements of a matrix. m Let m represent the coefficient of the m-th reflecting element in the active RIS, where m = 1, 2, ..., M. "|·|" is the modulo operator, ξ m Let |ξ| represent the amplification of the m-th reflecting element of the active RIS. m |≤p max p max This represents the maximum amplification power of the reflective element in the active RIS, where e is the natural constant, j is the imaginary representation, and v m Let f represent the phase of the m-th reflecting element of the active RIS, F represent the channel from the base station to the active RIS, and h represent the phase of the m-th reflecting element of the active RIS. B Let P represent the channel from the base station to the legitimate user, x represent the signal transmitted by the base station, x = ws, s represent the communication information flow between the base station and the legitimate user, s follows a complex Gaussian distribution with mean 0 and variance 1, w represent the transmit beamforming vector of the base station, and P J The h represents the firing power of the enemy target.JR h represents the channel from the enemy target to the active RIS. JU n represents the channel from the enemy target to the legitimate user. I This represents the thermal noise generated by the active RIS, with a mean of 0 and a covariance matrix of... I M Let n represent an M-order identity matrix. u This represents the received noise of a legitimate user, with a mean of 0 and a variance of .

[0016] Step 2.1b: According to y u The signal-to-interference-plus-noise ratio (SINR) of legitimate users is obtained. u , This leads to the achievable communication rate R for legitimate users. u R u =log2(1+SINR) u ), where "||·||2" is the 2-norm operator.

[0017] In step 2, the process of obtaining the eavesdropping rate by the illegal eavesdropper is as follows:

[0018] Step 2.2a: Assuming the eavesdropper can eliminate the interference signals sent by the enemy target, then the eavesdropper's received signal y... e Represented as Among them, g R This indicates the channel from the active RIS to the unauthorized eavesdropper, g B n represents the channel from the base station to the illegal eavesdropper. e This represents the received noise from an illegal eavesdropper, with a mean of 0 and a variance of .

[0019] Step 2.2b: According to y e The signal-to-interference-plus-noise ratio (SINR) obtained from the illegal eavesdropper e , This leads to the eavesdropping rate R of the illegal eavesdropper. e R e =log2(1+SINR) e ).

[0020] In step 2, the beam gain B(θ) generated in the direction of the active RIS toward the enemy target is expressed as: Where θ represents the angle of the enemy target relative to the active RIS, a(θ) represents the steering vector of the active RIS end, and a(θ) = [1, e] jπsin(θ) ,...,e jπ(M-1)sin(θ) ] T The superscript "T" indicates the transpose operation.

[0021] In step 2, the total system power consumption P total Represented as P total =P BS +P RIS , where P BS This indicates the power consumed by the base station. P RIS This indicates the power consumed by the active RIS. "||·|| F " is the F-norm operator, P SW P represents the power consumed in phase conversion and control. DC This indicates the power of a DC circuit.

[0022] h B h R And F is modeled as a Ricean channel model. in, This represents the line-of-sight link from the base station to the legitimate user. This refers to the non-line-of-sight link between the base station and the legitimate user. This indicates the line-of-sight link from the active RIS to the legitimate user. This indicates a non-line-of-sight link from the active RIS to the legitimate user. and All follow a complex Gaussian distribution with mean 0 and variance 1, β B This represents the path loss from the base station to the legitimate user. C0 represents the path loss at the reference distance d0, where d B α represents the path distance from the base station to the legitimate user. B κ represents the path loss exponent from the base station to the legitimate user. B β represents the Rice factor from the base station to the legitimate user. R This represents the path loss from the active RIS to the legitimate user. d R α represents the path distance from the active RIS to the legitimate user. R κ represents the path loss exponent from the active RIS to the legitimate user. R F represents the Rice factor for active RIS to legitimate users. LoS F represents the line-of-sight link from the base station to the active RIS. NLoS F represents the non-line-of-sight link from the base station to the active RIS. NLoS It follows a complex Gaussian distribution with mean 0 and variance 1, β F This represents the path loss from the base station to the active RIS. d F α represents the path distance from the base station to the active RIS. F κ represents the path loss exponent from the base station to the active RIS. FRepresents the Rice factor from the base station to the active RIS;

[0023] in, express The estimated value, express The estimation error, express The upper bound of the estimation error, δ represents the percentage of error. G represents i The estimated value, Δg i G represents i The estimation error, ||Δg i ||2≤∈ i ,∈ i G represents i The upper bound of the estimation error, and Modeled as a Ricean channel model, When JR This indicates the line-of-sight link from the enemy target to the active RIS. This indicates the non-line-of-sight link from the enemy target to the active RIS. It follows a complex Gaussian distribution with mean 0 and variance 1, β JR This represents the path loss from the enemy target to the active RIS. d JR α represents the path distance from the enemy target to the active RIS. JR κ represents the path loss exponent from the enemy target to the active RIS. JR This represents the Rice factor from the enemy target to the active RIS. When it is JU This represents the line-of-sight link between the enemy target and the legitimate user. This refers to the non-line-of-sight link between the enemy target and the legitimate user. It follows a complex Gaussian distribution with mean 0 and variance 1, β JU This indicates the path loss from the enemy target to the legitimate user. d JU α represents the path distance from the enemy target to the legitimate user. JU κ represents the path loss exponent from the enemy target to the legitimate user. JU Represents the Rice factor from the enemy target to the legitimate user, where i is B. This indicates the line-of-sight link between the base station and the illegal eavesdropper. This refers to the non-line-of-sight link between the base station and the illegal eavesdropper. It follows a complex Gaussian distribution with mean 0 and variance 1. This indicates the path loss from the base station to the illegal eavesdropper. This indicates the path distance from the base station to the illegal eavesdropper. This represents the path loss index from the base station to the illegal eavesdropper. Represents the Rice factor from the base station to the illegal eavesdropper, where i is R. This indicates an active RIS (Remote Information Sharing) link to the illegal eavesdropper. This indicates a non-line-of-sight link between an active RIS and an unauthorized eavesdropper. It follows a complex Gaussian distribution with mean 0 and variance 1. This indicates the path loss from the active RIS to the illegal eavesdropper. This indicates the path distance from the active RIS to the illegal eavesdropper. This represents the path loss index from the active RIS to the illegal eavesdropper. This indicates a Rice factor that actively targets illegal eavesdroppers via RIS.

[0024] In step 3, the optimization problem is described as follows:

[0025]

[0026] C2:R u ≥γ u ,

[0027] C3:R e ≤γ e ,

[0028] C4:|φ m |≤p max ,m=1,...,M.

[0029] in, γ represents the estimated value of θ, Δθ represents the estimation error of θ, and γ t γ represents the beam gain threshold. u γ represents the achievable communication rate threshold. e This indicates the threshold for the rate of eavesdropping.

[0030] In step 4, sub-problem 1 is described as follows: Subproblem 2 is described as follows:

[0031] stC1,C2,C3,C4

[0032] In step 4, the specific process of transforming subproblem 1 into convex problem 1 is as follows:

[0033] Step 4.1a: Let Transform the objective function of subproblem 1 into Where Tr(·) represents finding the trace of the matrix;

[0034] Rewrite constraint C1 of subproblem 1 as constraint C1':

[0035] make Rewrite constraint C2 of subproblem 1 as constraint C2': in, Then, define the cascaded channel h from the enemy target to the active RIS to the legitimate user. JRU for Its uncertainty is modeled as This leads to a joint channel connecting adversary targets to legitimate users. Represented as in, h JRU The estimated value, Δh JRU h JRU The estimation error, express The estimated value, express The estimation error, Next, define and combined Transform the left side of constraint C2' into The constraint C2 is obtained, where φ represents the column vector consisting of the coefficients of all reflective elements of the active RIS, φ = [φ1, φ2, ..., φ]. M ] T Re{·} denotes taking the real part; then, using the S lemma, constraint C2” is transformed into the form of a linear matrix inequality, thus obtaining constraint C2”': Where λ represents a non-negative auxiliary variable, I M+1 Describes an identity matrix of order M+1.

[0036] Define the cascaded channel G from the base station to the active RIS to the illegal eavesdropper. BRE for Its uncertainty is modeled as The joint channel G from the base station to the illegal eavesdropper is then represented as: in, G represents BRE The estimated value, ΔG BRE G represents BRE The estimation error, Let G be the estimated value, ΔG be the estimation error of G, and ||ΔG||2≤∈G , Then, constraint C3 of subproblem 1 is transformed into constraint C3': in, Next, based on the properties of finding the trace of a matrix: Tr(X H Y)=vec H Given (X)vec(Y), we have: Where vec(·) represents the vectorization operation, This is the Kroc inner product operator, g = vec(ΔG) H ), Then transform constraint C3' into constraint C3”: Then, using Lemma S, constraint C3” is transformed into a linear matrix inequality, thus obtaining constraint C3”’: Where μ represents a non-negative auxiliary variable, I (M+1)N Let represent the identity matrix of order (M+1)N;

[0037] Step 4.1b: Describe convex problem 1 as:

[0038] The specific process of transforming subproblem 2 into convex problem 2 is as follows:

[0039] Step 4.2a: Introduce the auxiliary variable η to transform subproblem 2 into:

[0040]

[0041] C1,C2,C3,C4

[0042] Step 4.2b: Rewrite constraint C5 as constraint C5': in, Then Substituting into constraint C5', we get This leads to the constraint C5”: Next, using Lemma S, constraint C5” is transformed into a linear matrix inequality, thus obtaining the constraint. in, Represents a non-negative auxiliary variable;

[0043] Transform constraint C4 into constraint C4': in, express The m-th element;

[0044] Step 4.2c: Describe convex problem 2 as: in, I MLet represent an M-order identity matrix, with the superscript "*" indicating conjugate operation. Indicates by The submatrix formed by rows 1 to M and columns 1 to M.

[0045] Compared with the prior art, the advantages of the present invention are as follows:

[0046] 1) This invention provides a beamforming method that minimizes total system power consumption by jointly optimizing the transmit beamforming of a dual-function base station and the phase shift matrix of an active reconfigurable smart surface, achieving green and energy-efficient communication. Compared to traditional methods, this invention addresses issues such as communication link obstruction and limited sensing coverage by introducing an active reconfigurable smart surface. Compared to traditional passive reconfigurable smart surfaces, the phase shift matrix optimization method for active reconfigurable smart surfaces proposed in this invention further reduces system power consumption. This invention effectively solves the variable coupling problem using an alternating optimization algorithm and utilizes matrix theory methods such as the S-lemma to address practical application scenarios where illegal node channel state information cannot be perfectly obtained. Finally, the optimization problem is derived into a convex form for solution. This beamforming method provides an efficient and environmentally friendly solution for modern wireless communication systems, contributing to more stable and efficient communication services in complex electromagnetic environments such as cities and campuses.

[0047] 2) This invention specifically addresses a challenging integrated communication and sensing application scenario where the system must perform both user communication and target sensing tasks in a complex environment where both illegal eavesdroppers and malicious jammers coexist. By establishing communication performance indicators, sensing performance indicators, and power consumption performance indicators for the system model, a problem is formulated to minimize the total system power consumption. While formulating the optimization problem, the transmission rate of communication users is ensured to meet requirements, and their data is not illegally obtained by illegal eavesdroppers. Furthermore, this invention considers the imperfections in channel state information related to illegal nodes; the designed beamforming method exhibits robustness even when channel acquisition errors exist. The system also possesses effective anti-interference capabilities, resisting interference signals from malicious jammers, thereby achieving coordinated optimization of communication, sensing, and physical layer security. Attached Figure Description

[0048] Figure 1 This is a flowchart illustrating the overall implementation process of the method of the present invention;

[0049] Figure 2 This is a simplified schematic diagram of the active RIS-ISAC system model in the method of this invention;

[0050] Figure 3 The present invention method and the comparative method are based on the communication achievable rate threshold γ. uA comparative diagram showing the changes in total system power consumption as the system changes.

[0051] Figure 4 To achieve the communication achievable rate threshold γ u At a transmission power P of 4 bps / Hz, the method of this invention and the comparative method differ in that the jammer's transmission power P... J A comparative curve diagram showing the change in total system power consumption during the change. Detailed Implementation

[0052] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0053] The present invention proposes an active RIS-ISAC system beamforming method for preventing eavesdropping and interference, the overall implementation flowchart of which is shown below. Figure 1 As shown, it includes the following steps:

[0054] Step 1: Establish an active RIS-ISAC (Reconfigurable Smart Surface-Assisted Communication and Sensing Integration) system model, such as... Figure 2 As shown, the model includes a base station (BS) with dual radar and communication functions, an active RIS, a legitimate user, an illegal eavesdropper (Eve), and an enemy target, i.e., a jammer (Jam / Tar). With the assistance of the active RIS, the base station communicates with the legitimate user via downlink. The illegal eavesdropper attempts to steal the signals sent by the base station to the legitimate user. The base station detects the enemy target, and the enemy target actively sends jamming signals to the legitimate user to disrupt the communication between the base station and the legitimate user.

[0055] Further specified, the base station is equipped with a uniform linear array with N antennas, which is N=12 in this embodiment; the number of reflective elements of the active RIS is M, which is M=30 in this embodiment; both legitimate users and illegal eavesdroppers are equipped with a single antenna.

[0056] Step 2: Establish the communication performance indicators, sensing performance indicators, and power consumption performance indicators of the active RIS-ISAC system model. The communication performance indicators include the reachable communication rate of legitimate users and the eavesdropping rate of illegal eavesdroppers. The sensing performance indicators include the beam gain generated by the active RIS in the direction to the enemy target. The power consumption performance indicators include the total power consumption of the system.

[0057] In this embodiment, the process of obtaining the reachable rate of a legitimate user in step 2 is as follows:

[0058] Step 2.1a: Transfer the received signal y from the legitimate user u Represented as Wherein, the superscript "H" indicates the conjugate transpose operation, h RThis indicates a channel for proactive RIS to legitimate users. Φ represents the phase shift matrix of the active RIS. As a defined symbol, diag(·) is used to extract or construct the diagonal elements of a matrix, φ m Let m represent the coefficient of the m-th reflecting element in the active RIS, where m = 1, 2, ..., M. "|·|" is the modulo operator, ξ m Let |ξ| represent the amplification of the m-th reflecting element of the active RIS. m |≤p max p max This represents the maximum amplification power of the reflective element of the active RIS, which is taken as p in this embodiment. max =40dB, e is the natural constant, e = 2.7…, j is the imaginary number representation, v m Let f represent the phase of the m-th reflecting element of the active RIS, and F represent the channel from the base station to the active RIS. h B This indicates the channel from the base station to the legitimate user. x represents the signal transmitted by the base station, x = ws, s represents the communication information flow between the base station and the legitimate user, and s follows a complex Gaussian distribution with mean 0 and variance 1, i.e. Let represent a complex Gaussian distribution with mean 0 and variance 1, and w represent the transmit beamforming vector of the base station. P J The h represents the firing power of the enemy target. JR This indicates the channel from the enemy target to the active RIS. h JU This represents the channel from the enemy target to the legitimate user. n I This represents the thermal noise generated by the active RIS, with a mean of 0 and a covariance matrix of... I M Let n represent an M-order identity matrix. u This represents the received noise of a legitimate user, with a mean of 0 and a variance of . In this embodiment, take

[0059] Step 2.1b: According to y u The signal-to-interference-plus-noise ratio (SINR) of legitimate users is obtained. u , This leads to the achievable communication rate R for legitimate users. u R u =log2(1+SINR) u ), where "||·||2" is the 2-norm operator.

[0060] In this embodiment, the process of obtaining the eavesdropping rate of the illegal eavesdropper in step 2 is as follows:

[0061] Step 2.2a: Since there is cooperation between the illegal eavesdropper and the enemy target, it can be assumed that the illegal eavesdropper can eliminate the interference signals sent by the enemy target. Therefore, the illegal eavesdropper's received signal y... e Represented as Among them, g R This indicates an active RIS signal to the channel of an unauthorized eavesdropper. g B This indicates the channel from the base station to the illegal eavesdropper. n e This represents the received noise from an illegal eavesdropper, with a mean of 0 and a variance of . In this embodiment, take

[0062] Step 2.2b: According to y e The signal-to-interference-plus-noise ratio (SINR) obtained from the illegal eavesdropper e , This leads to the eavesdropping rate R of the illegal eavesdropper. e R e =log2(1+SINR) e ).

[0063] In this embodiment, in step 2, it is assumed that the direct link from the base station to the enemy target (jammer) is completely blocked by buildings, trees, etc. In this case, deploying an active RIS can create a virtual line-of-sight link between the base station and the enemy target to achieve the purpose of enemy target perception. Here, the beam gain generated by the active RIS on the enemy target is used as an indicator to measure the perception performance. The beam gain B(θ) generated by the active RIS in the direction to the enemy target is expressed as: Where θ represents the angle of the enemy target relative to the active RIS, a(θ) represents the steering vector of the active RIS end, and a(θ) = [1, e] jπsin(θ) ,...,e jπ(M-1)sin(θ) ] T The superscript "T" indicates the transpose operation. In practical systems, the angle θ of the enemy target relative to the active RIS is uncertain. This invention assumes that the true position of the enemy target is limited to a certain range, i.e. in, The value of θ is represented by Δθ, and the estimation error of θ is represented by Δθ. In this embodiment, Δθ is taken as Δθ. Δθ = 5°.

[0064] In this embodiment, in step 2, unlike the passive RIS, the active RIS is equipped with a set of reflective elements capable of amplifying the signal. Its power consumption is non-negligible for the entire system. Therefore, the total system power consumption should be equal to the sum of the power consumption of the base station and the active RIS. The total system power consumption P... total Represented as P total =P BS +P RIS , where P BS This indicates the power consumed by the base station. P RIS This indicates the power consumed by the active RIS. "‖·‖ F " is the F-norm operator, P SW P represents the power consumed in phase conversion and control. DC This represents the power of the DC circuit; in this embodiment, P is taken as the power. SW = -10dBm, P DC = -5dBm.

[0065] In this specific embodiment, h B h R And F is modeled as a Ricean channel model. F is determined by both large-scale fading and small-scale fading. in, This represents the line-of-sight link from the base station to the legitimate user. This refers to the non-line-of-sight link between the base station and the legitimate user. This indicates the line-of-sight link from the active RIS to the legitimate user. This indicates a non-line-of-sight link from the active RIS to the legitimate user. and All follow a complex Gaussian distribution with mean 0 and variance 1, β B This represents the path loss from the base station to the legitimate user. C0 represents the path loss at the reference distance d0. In this embodiment, C0 = -30dB and d0 = 1 meter. B α represents the path distance from the base station to the legitimate user. B κ represents the path loss exponent from the base station to the legitimate user. B The Rice factor represents the distance from the base station to the legitimate user. and The intensity ratio, β R This represents the path loss from the active RIS to the legitimate user. d R α represents the path distance from the active RIS to the legitimate user. R κ represents the path loss exponent from the active RIS to the legitimate user. RThe Rice factor represents the active RIS to legitimate users, and its representation is... and The strength ratio, in this embodiment, is taken as κ. B =0, α B =3.5, κ R =1, α R =2.2, F LoS F represents the line-of-sight link from the base station to the active RIS. NLoS F represents the non-line-of-sight link from the base station to the active RIS. NLoS It follows a complex Gaussian distribution with mean 0 and variance 1, β F This represents the path loss from the base station to the active RIS. d F α represents the path distance from the base station to the active RIS. F κ represents the path loss exponent from the base station to the active RIS. F The Rice factor representing the distance from the base station to the active RIS is F. LoS With F NLoS The strength ratio, in this embodiment, is taken as κ. F =1, α F =2.2.

[0066] For legitimate systems, there is a certain uncertainty regarding illegitimate nodes (illegal eavesdroppers and adversaries), and their associated channels are difficult to obtain perfectly in practice. Therefore, this invention employs a bounded channel error model to describe this uncertainty. in, express The estimated value, express The estimation error, express The upper bound of the estimation error, δ represents the percentage of error; in this specific embodiment, δ = 0.01. G represents i The estimated value, Δg i G represents i The estimation error, ||Δg i ||2≤∈ i ,∈ i G represents i The upper bound of the estimation error, and Modeled as a Ricean channel model, When JR This indicates the line-of-sight link from the enemy target to the active RIS. This indicates the non-line-of-sight link from the enemy target to the active RIS. It follows a complex Gaussian distribution with mean 0 and variance 1, β JR This represents the path loss from the enemy target to the active RIS. d JR α represents the path distance from the enemy target to the active RIS. JR κ represents the path loss exponent from the enemy target to the active RIS. JR Represents the Rice factor from the enemy target to the active RIS, which represents and Strength ratio, When it is JU This represents the line-of-sight link between the enemy target and the legitimate user. This refers to the non-line-of-sight link between the enemy target and the legitimate user. It follows a complex Gaussian distribution with mean 0 and variance 1, β JU This indicates the path loss from the enemy target to the legitimate user. d JU α represents the path distance from the enemy target to the legitimate user. JU κ represents the path loss exponent from the enemy target to the legitimate user. JU The Rice factor represents the distance from the enemy target to the legitimate user, and its representative is... and The strength ratio, in this embodiment, is taken as κ. JR =1, α JR =2.2、κ JU =0, α JU =3.5, when i is B This indicates the line-of-sight link between the base station and the illegal eavesdropper. This refers to the non-line-of-sight link between the base station and the illegal eavesdropper. It follows a complex Gaussian distribution with mean 0 and variance 1. This indicates the path loss from the base station to the illegal eavesdropper. This indicates the path distance from the base station to the illegal eavesdropper. This represents the path loss index from the base station to the illegal eavesdropper. The Rice factor represents the connection between the base station and the illegal eavesdropper. and The strength ratio, when i is R This indicates an active RIS (Remote Information Sharing) link to the illegal eavesdropper. This indicates a non-line-of-sight link between an active RIS and an unauthorized eavesdropper. It follows a complex Gaussian distribution with mean 0 and variance 1. This indicates the path loss from the active RIS to the illegal eavesdropper. This indicates the path distance from the active RIS to the illegal eavesdropper. This represents the path loss index from the active RIS to the illegal eavesdropper. This indicates the Rice factor, representing the active RIS to illegal eavesdroppers. and The strength ratio, in this embodiment, is taken as...

[0067] Step 3: With the goal of minimizing the total power consumption of the system, and constrained by the reachable communication rate of legitimate users, the eavesdropping rate of illegal eavesdroppers, and the beam gain generated in the direction of the active RIS to the enemy target, jointly optimize the base station's transmit beamforming vector and the phase shift matrix of the active RIS to construct an optimization problem.

[0068] To further refine the problem, the optimization problem can be described as follows:

[0069]

[0070] C2:R u ≥γ u ,

[0071] C3:R e ≤γ e ,

[0072] C4:|φ m |≤p max ,m=1,...,M.

[0073] Where min is the function for finding the minimum value, st means "constrained by...", and γ t γ represents the beam gain threshold. u γ represents the achievable communication rate threshold. e In this embodiment, γ represents the eavesdropping rate threshold. t =20dB, γ u =5bps / Hz, γ e =1bps / Hz, constraint C1 indicates that the desired beam gain generated by the active RIS in the direction toward the enemy target is not less than γ. t Constraint C2 indicates that the reachable communication rate of a legitimate user is not less than γ. u Constraint C3 indicates that the eavesdropping rate of the illegal eavesdropper is no greater than γ. e Constraint C4 represents the coefficient constraint for each reflective element of the active RIS.

[0074] Step 4: Since the optimization problem constructed in Step 3 is a non-convex variable-coupled optimization problem, which is difficult to solve, the optimization problem is split into subproblem 1 and subproblem 2. Subproblem 1 is used to optimize the transmit beamforming vector of the base station when the phase shift matrix of the active RIS is fixed, and subproblem 2 is used to optimize the phase shift matrix of the active RIS when the transmit beamforming vector of the base station is fixed. Then, subproblem 1 is transformed into convex problem 1, and subproblem 2 is transformed into convex problem 2.

[0075] To further refine, subproblem 1 is described as follows: Subproblem 2 is described as follows:

[0076] stC1,C2,C3,C4

[0077] Further specifying the process, the specific steps to transform subproblem 1 into convex problem 1 are as follows:

[0078] Step 4.1a: Let Transform the objective function of subproblem 1 into Here, Tr(·) represents finding the trace of a matrix.

[0079] Rewrite constraint C1 of subproblem 1 as constraint C1':

[0080] make Rewrite constraint C2 of subproblem 1 as constraint C2': in, Then, define the cascaded channel h from the enemy target to the active RIS to the legitimate user. JRU for Its uncertainty can be modeled as This leads to a joint channel connecting adversary targets to legitimate users. (Including cascaded channels and direct channels) is represented as in, h JRU The estimated value, Δh JRU h JRU The estimation error, express The estimated value, express The estimation error, Next, define and combined Transform the left side of constraint C2' into The constraint C2 is obtained, where φ represents the column vector consisting of the coefficients of all reflective elements of the active RIS, φ = [φ1, φ2, ..., φ]. M ] TRe{·} denotes taking the real part; due to the introduction of the bounded channel error model, constraint C2” consists of infinitely many non-convex constraints. Therefore, to solve this problem, the S lemma is then used to transform constraint C2” into the form of a linear matrix inequality, thus obtaining constraint C2”': Where λ represents a non-negative auxiliary variable, I M+1 Describes an identity matrix of order M+1.

[0081] To transform constraint C3 into a convex form, we define the cascaded channel G from the base station to the active RIS to the illegal eavesdropper. BRE for Its uncertainty can be modeled as The joint channel G (including direct channels and cascaded channels) from the base station to the illegal eavesdropper is then represented as: in, G represents BRE The estimated value, ΔG BRE G represents BRE The estimation error, Let G be the estimated value, ΔG be the estimation error of G, and ||ΔG||2≤∈ G , Then, constraint C3 of subproblem 1 is transformed into constraint C3': in, Next, based on the properties of finding the trace of a matrix: Tr(X H Y)=vec H Given (X)vec(Y), we have: Where vec(·) represents the vectorization operation, This is the Kroc inner product operator, g = vec(ΔG) H ), Then transform constraint C3' into constraint C3”: Then, using Lemma S, constraint C3” is transformed into a linear matrix inequality, thus obtaining constraint C3”’: Where μ represents a non-negative auxiliary variable, I (M+1)N Let represent an identity matrix of order (M+1)N.

[0082] Step 4.1b: Describe convex problem 1 as:

[0083] The specific process of transforming subproblem 2 into convex problem 2 is as follows:

[0084] Step 4.2a: Introduce the auxiliary variable η to transform subproblem 2 into:

[0085]

[0086] C1,C2,C3,C4

[0087] Step 4.2b: Rewrite constraint C5 as constraint C5': in, Then Substituting into constraint C5', we get This leads to the constraint C5”: Next, using Lemma S, constraint C5” is transformed into a linear matrix inequality, thus obtaining constraint C5”': in, This represents a non-negative auxiliary variable.

[0088] Transform constraint C4 into constraint C4': in, express The m-th element.

[0089] Step 4.2c: Describe convex problem 2 as: in, I M Let represent an M-order identity matrix, with the superscript "*" indicating conjugate operation. Indicates by The submatrix formed by rows 1 to M and columns 1 to M.

[0090] Step 5: Use the alternating optimization algorithm to iteratively solve convex problem 1 and convex problem 2 to obtain the optimal solutions for the base station's transmit beamforming vector and the phase shift matrix of the active RIS.

[0091] Here, the alternating optimization algorithm is a commonly used algorithm to iteratively solve for multiple optimization variables. Iteration stops when the iteration accuracy reaches a preset accuracy threshold or the number of iterations reaches a preset maximum number of iterations, thus obtaining the optimal solution for each optimization variable. In this embodiment, the preset accuracy threshold can be set to 10. -3 The maximum number of iterations can be preset to 30. During each iteration, a convex optimization solver such as CVX is used to solve convex problem 1 and convex problem 2.

[0092] The following simulations further illustrate the feasibility and effectiveness of the method of the present invention.

[0093] In the simulation, the comparison methods included the random RIS phase shift method (which does not optimize the phase shift matrix of the active RIS and is given randomly) and the passive RIS method (the passive RIS can only adjust the signal phase and does not have the ability to amplify the signal). The performance curves of the method of the present invention were compared under the condition of perfect CSI (assuming that the channel state information of illegal nodes (illegal eavesdroppers and enemy targets) can be perfectly obtained).

[0094] Figure 3The method of this invention (with an error percentage of 0.01) is presented, along with the method of this invention under perfect CSI conditions and the comparative method under the communication achievable rate threshold γ. u A comparison curve showing the change in total system power consumption as the system changes. From Figure 3 As can be seen, when the demand for communication continues to increase, that is, when the communication achievable rate threshold γ is reached... u As the CSI increases, the total system power consumption tends to rise, but the total system power consumption of the method of this invention remains at a low level, especially under perfect CSI conditions, where the total system power consumption is basically maintained below 10W. Furthermore, the method of this invention is superior to passive RIS methods and random RIS phase-shift methods in reducing total system power consumption, especially when the communication achievable rate threshold γ is reached. u As the power consumption increases, the difference between the total system power consumption of the method of the present invention and the comparative method also increases, which verifies the effectiveness of the method of the present invention in green and energy-saving communication.

[0095] Figure 4 The threshold γ for achievable communication rate is given. u At 4 bps / Hz, the method of this invention (error percentage of 0.01) and the method of this invention under perfect CSI conditions, as well as the comparison method, under the jammer's transmit power P... J A comparison curve showing the change in total system power consumption as the system changes. From Figure 4 As can be seen, the total system power consumption increases with the increase of the jammer's power, and the total system power consumption of the method of this invention is lower than that of the passive RIS method and the random RIS phase-shift method. Compared with the passive RIS-assisted ISAC system, the active RIS can not only adjust the phase of the incident signal, but also has the ability to amplify the signal. Even though its reflective element itself will cause some power consumption, the deployment of active RIS has a huge improvement on the overall system performance. Compared with the random RIS phase-shift method, the RIS phase-shift optimization method proposed in this invention is very effective, which can greatly reduce the total system power consumption and achieve the effect of green and energy-saving communication while ensuring secure anti-jamming transmission and target perception.

Claims

1. A beamforming method for an active RIS-ISAC system to prevent eavesdropping and interference, characterized in that... Includes the following steps: Step 1: Establish an active RIS-ISAC system model. This model includes a base station with dual radar and communication functions, an active RIS, a legitimate user, an illegal eavesdropper, and an enemy target, i.e., a jammer. With the assistance of the active RIS, the base station communicates with the legitimate user via downlink. The illegal eavesdropper attempts to steal the signal sent by the base station to the legitimate user. The base station detects the enemy target, and the enemy target will actively send jamming signals to the legitimate user to interfere with the legitimate user and destroy the communication between the base station and the legitimate user. Step 2: Establish the communication performance indicators, sensing performance indicators, and power consumption performance indicators of the active RIS-ISAC system model. The communication performance indicators include the reachable communication rate of legitimate users and the eavesdropping rate of illegal eavesdroppers. The sensing performance indicators include the beam gain generated by the active RIS in the direction to the enemy target. The power consumption performance indicators include the total power consumption of the system. The total power consumption of the system includes the power consumed by the base station and the active RIS. The channels from the enemy target to the active RIS and from the enemy target to the legitimate user used in the process of obtaining the communication reachability of the legitimate user, as well as the channels from the active RIS to the illegal eavesdropper and from the base station to the illegal eavesdropper used in the process of obtaining the eavesdropping rate of the illegal eavesdropper, are all modeled by adding an estimated value to the estimation error with an upper bound. Step 3: With the goal of minimizing the total power consumption of the system, and constrained by the reachable communication rate of legitimate users, the eavesdropping rate of illegal eavesdroppers, and the beam gain generated in the direction of the active RIS to the enemy target, jointly optimize the base station's transmit beamforming vector and the phase shift matrix of the active RIS to construct an optimization problem; Step 4: Decompose the optimization problem into subproblem 1 and subproblem 2. Subproblem 1 is used to optimize the base station's transmit beamforming vector with a fixed phase shift matrix of the active RIS. Subproblem 2 is used to optimize the active RIS's phase shift matrix with a fixed transmit beamforming vector of the base station. Then, subproblem 1 is transformed into convex problem 1, and subproblem 2 is transformed into convex problem 2. The S-lemma is used in the constraint transformation of the reachable communication rate of legitimate users and the eavesdropping rate of illegitimate eavesdroppers. Step 5: Use the alternating optimization algorithm to iteratively solve convex problem 1 and convex problem 2 to obtain the optimal solutions for the base station's transmit beamforming vector and the phase shift matrix of the active RIS.

2. The beamforming method for an active RIS-ISAC system to prevent eavesdropping and interference according to claim 1, characterized in that... In step 1, the base station is equipped with a uniform linear array with N antennas; the number of reflective elements of the active RIS is M. Both legitimate users and illegal eavesdroppers are equipped with a single antenna.

3. The beamforming method for an active RIS-ISAC system to prevent eavesdropping and interference according to claim 2, characterized in that... In step 2, the process of obtaining the reachable rate of a legitimate user is as follows: Step 2.1a: Transfer the received signal y from the legitimate user u Represented as Wherein, the superscript "H" indicates the conjugate transpose operation, h R Let Φ represent the channel from the active RIS to the legitimate user, and let Φ represent the phase shift matrix of the active RIS. The `diag(·)` function is used to extract or construct the diagonal elements of a matrix. m This represents the coefficient of the m-th reflecting element in the active RIS. "|·|" is the modulo operator, ξ m Let |ξ| represent the amplification of the m-th reflecting element of the active RIS. m |≤p max p max This represents the maximum amplification power of the reflective element in the active RIS, where e is the natural constant, j is the imaginary representation, and v m Let f represent the phase of the m-th reflecting element of the active RIS, F represent the channel from the base station to the active RIS, and h represent the phase of the m-th reflecting element of the active RIS. B Let P represent the channel from the base station to the legitimate user, x represent the signal transmitted by the base station, x = ws, s represent the communication information flow between the base station and the legitimate user, s follows a complex Gaussian distribution with mean 0 and variance 1, w represent the transmit beamforming vector of the base station, and P J The h represents the firing power of the enemy target. JR h represents the channel from the enemy target to the active RIS. JU n represents the channel from the enemy target to the legitimate user. I This represents the thermal noise generated by the active RIS, with a mean of 0 and a covariance matrix of... I M Let n represent an M-order identity matrix. u This represents the received noise of a legitimate user, with a mean of 0 and a variance of . Step 2.1b: According to y u The signal-to-interference-plus-noise ratio (SINR) of legitimate users is obtained. u , This leads to the achievable communication rate R for legitimate users. u R u =log2(1+SINR) u ), where "||·||2" is the 2-norm operator.

4. The beamforming method for an active RIS-ISAC system to prevent eavesdropping and interference according to claim 3, characterized in that... In step 2, the process of obtaining the eavesdropping rate by the illegal eavesdropper is as follows: Step 2.2a: Assuming the eavesdropper can eliminate the interference signals sent by the enemy target, then the eavesdropper's received signal y... e Represented as Among them, g R This indicates the channel from the active RIS to the unauthorized eavesdropper, g B n represents the channel from the base station to the illegal eavesdropper. e This represents the received noise from an illegal eavesdropper, with a mean of 0 and a variance of . Step 2.2b: According to y e The signal-to-interference-plus-noise ratio (SINR) obtained from the illegal eavesdropper e , This leads to the eavesdropping rate R of the illegal eavesdropper. e R e =log2(1+SINR) e ).

5. The beamforming method for an active RIS-ISAC system to prevent eavesdropping and interference according to claim 4, characterized in that... In step 2, the beam gain B(θ) generated in the direction of the active RIS toward the enemy target is expressed as: Where θ represents the angle of the enemy target relative to the active RIS, a(θ) represents the steering vector of the active RIS end, and a(θ) = [1, e] jπsin(θ) ,...,e jπ(M-1)sin(θ) ] T The superscript "T" indicates the transpose operation.

6. The beamforming method for an active RIS-ISAC system to prevent eavesdropping and interference according to claim 5, characterized in that... In step 2, the total system power consumption P total Represented as P total =P BS +P RIS , where P BS This indicates the power consumed by the base station. P RIS This indicates the power consumed by the active RIS. ||·|| F " is the F-norm operator, P SW P represents the power consumed in phase conversion and control. DC This indicates the power of a DC circuit.

7. The beamforming method for an active RIS-ISAC system to prevent eavesdropping and interference as described in claim 6, characterized in that... h B h R And F is modeled as a Ricean channel model. in, This represents the line-of-sight link from the base station to the legitimate user. This refers to the non-line-of-sight link between the base station and the legitimate user. This indicates the line-of-sight link from the active RIS to the legitimate user. This indicates a non-line-of-sight link from the active RIS to the legitimate user. and All follow a complex Gaussian distribution with mean 0 and variance 1, β B This represents the path loss from the base station to the legitimate user. C0 represents the path loss at the reference distance d0, where d B α represents the path distance from the base station to the legitimate user. B κ represents the path loss exponent from the base station to the legitimate user. B β represents the Rice factor from the base station to the legitimate user. R This represents the path loss from the active RIS to the legitimate user. d R α represents the path distance from the active RIS to the legitimate user. R κ represents the path loss exponent from the active RIS to the legitimate user. R F represents the Rice factor for active RIS to legitimate users. LoS F represents the line-of-sight link from the base station to the active RIS. NLoS F represents the non-line-of-sight link from the base station to the active RIS. NLoS It follows a complex Gaussian distribution with mean 0 and variance 1, β F This represents the path loss from the base station to the active RIS. d F α represents the path distance from the base station to the active RIS. F κ represents the path loss exponent from the base station to the active RIS. F Represents the Rice factor from the base station to the active RIS; in, express The estimated value, express The estimation error, express The upper bound of the estimation error, δ represents the percentage of error. G represents i The estimated value, Δg i G represents i The estimation error, ||Δg i ||2≤∈ i ,∈ i G represents i The upper bound of the estimation error, and Modeled as a Ricean channel model, When JR This indicates the line-of-sight link from the enemy target to the active RIS. This indicates the non-line-of-sight link from the enemy target to the active RIS. It follows a complex Gaussian distribution with mean 0 and variance 1, β JR This represents the path loss from the enemy target to the active RIS. d JR α represents the path distance from the enemy target to the active RIS. JR κ represents the path loss exponent from the enemy target to the active RIS. JR This represents the Rice factor from the enemy target to the active RIS. When it is JU This represents the line-of-sight link between the enemy target and the legitimate user. This refers to the non-line-of-sight link between the enemy target and the legitimate user. It follows a complex Gaussian distribution with mean 0 and variance 1, β JU This indicates the path loss from the enemy target to the legitimate user. d JU α represents the path distance from the enemy target to the legitimate user. JU κ represents the path loss exponent from the enemy target to the legitimate user. JU Represents the Rice factor from the enemy target to the legitimate user, where i is B. This indicates the line-of-sight link between the base station and the illegal eavesdropper. This refers to the non-line-of-sight link between the base station and the illegal eavesdropper. It follows a complex Gaussian distribution with mean 0 and variance 1. This indicates the path loss from the base station to the illegal eavesdropper. This indicates the path distance from the base station to the illegal eavesdropper. This represents the path loss index from the base station to the illegal eavesdropper. Represents the Rice factor from the base station to the illegal eavesdropper, where i is R. This indicates an active RIS (Remote Information Provider) line-of-sight link to the illegal eavesdropper. This indicates a non-line-of-sight link between an active RIS and an unauthorized eavesdropper. It follows a complex Gaussian distribution with mean 0 and variance 1. This indicates the path loss from the active RIS to the illegal eavesdropper. This indicates the path distance from the active RIS to the illegal eavesdropper. This represents the path loss index from the active RIS to the illegal eavesdropper. This indicates a Rice factor that actively targets illegal eavesdroppers via RIS.

8. The beamforming method for an active RIS-ISAC system to prevent eavesdropping and interference according to claim 7, characterized in that... In step 3, the optimization problem is described as follows: in, γ represents the estimated value of θ, Δθ represents the estimation error of θ, and γ t γ represents the beam gain threshold. u γ represents the achievable communication rate threshold. e This indicates the threshold for the rate of eavesdropping.

9. The beamforming method for an active RIS-ISAC system to prevent eavesdropping and interference as described in claim 8, characterized in that... In step 4, sub-problem 1 is described as follows: Subproblem 2 is described as follows:

10. The beamforming method for an active RIS-ISAC system to prevent eavesdropping and interference according to claim 9, characterized in that... In step 4, the specific process of transforming subproblem 1 into convex problem 1 is as follows: Step 4.1a: Let Transform the objective function of subproblem 1 into Where Tr(·) represents finding the trace of the matrix; Rewrite constraint C1 of subproblem 1 as constraint C1': make Rewrite constraint C2 of subproblem 1 as constraint C2': in, Then, define the cascaded channel h from the enemy target to the active RIS to the legitimate user. JRU for Its uncertainty is modeled as This leads to a joint channel connecting adversary targets to legitimate users. Represented as in, h JRU The estimated value, Δh JRU h JRU The estimation error, express The estimated value, express The estimation error, Next, define and combined Transform the left side of constraint C2' into The constraint C2 is obtained, where φ represents the column vector consisting of the coefficients of all reflective elements of the active RIS, φ = [φ1, φ2, ..., φ]. M ] T Re{·} denotes taking the real part; then, using the S lemma, constraint C2” is transformed into the form of a linear matrix inequality, thus obtaining constraint C2”': Where λ represents a non-negative auxiliary variable, I M+1 Describes an identity matrix of order M+1. Define the cascaded channel G from the base station to the active RIS to the illegal eavesdropper. BRE for Its uncertainty is modeled as The joint channel G from the base station to the illegal eavesdropper is then represented as: in, G represents BRE The estimated value, ΔG BRE G represents BRE The estimation error, Let G be the estimated value, ΔG be the estimation error of G, and ||ΔG||2≤∈ G , Then, constraint C3 of subproblem 1 is transformed into constraint C3': in, Next, based on the properties of finding the trace of a matrix: Tr(X H Y)=vec H Given (X)vec(Y), we have: Where vec(·) represents the vectorization operation, This is the Kroc inner product operator, g = vec(ΔG) H ), Then transform constraint C3' into constraint C3”: Then, using Lemma S, constraint C3” is transformed into a linear matrix inequality, thus obtaining constraint C3”’: Where μ represents a non-negative auxiliary variable, I (M+1)N Let represent the identity matrix of order (M+1)N; Step 4.1b: Describe convex problem 1 as: The specific process of transforming subproblem 2 into convex problem 2 is as follows: Step 4.2a: Introduce the auxiliary variable η to transform subproblem 2 into: Step 4.2b: Rewrite constraint C5 as constraint C5': in, Then Substituting into constraint C5', we get This leads to the constraint C5”: Next, using Lemma S, constraint C5” is transformed into a linear matrix inequality, thus obtaining constraint C5”': in, Represents a non-negative auxiliary variable; Transform constraint C4 into constraint C4': in, express The m-th element; Step 4.2c: Describe convex problem 2 as: in, I M Let represent an M-order identity matrix, with the superscript "*" indicating conjugate operation. Indicates by The submatrix formed by rows 1 to M and columns 1 to M.

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