A method for integrated wave control based on super-diagonal intelligent reflecting surface

Abstract

The application discloses a kind of based on super-diagonal intelligent reflector surface communication and sensing integrated joint waveform control method, it is related to super-diagonal intelligent reflector surface technical field, the method first obtains the nimble adjustment or stable convergence mode of communication system, for different modes respectively using Lagrange penalty function, Takagi decomposition's Riemann manifold gradient descent method, optimal base station transmitting beam matrix and optimal BD-RIS reflection phase shift matrix are solved, and optimal base station transmitting beam matrix and optimal BD-RIS reflection phase shift matrix are transmitted to base station radio frequency front end and BD-RIS impedance matching network, realize the accurate waveform control of communication and sensing integrated signal, effectively improve communication and sensing performance, avoid the numerical oscillation and super parameter sensitivity brought by double-loop of penalty function method, simplify algorithm solving structure, effectively approach the theoretical performance upper bound of system under the premise of strictly meeting physical constraint, applicable to complex dynamic communication and sensing integrated scene.

Description

Technical Field

[0001] This invention relates to the field of super-diagonal intelligent reflective surface technology, and more specifically to a combined waveform control method based on super-diagonal intelligent reflective surface with integrated sensing. Background Technology

[0002] Integrated Sensing and Communication (ISAC) is a core technology enabling future 6G wireless networks. To address complex non-line-of-sight fading and severe path loss in high-frequency bands, introducing reconfigurable intelligent surfaces (RIS) to intelligently reconfigure propagation channels has become a key approach to improving system performance. Traditional standard RIS can only independently adjust the reflection phase of each array element, resulting in a physical bottleneck in spatial control freedom. Therefore, the Beyond Diagonal Reconfigurable Intelligent Surface (BD-RIS), with its fully connected topology, has emerged. Benefiting from the non-diagonal scattering mechanism between array elements, BD-RIS can guide the dynamic routing and redistribution of signal energy among different array elements, greatly enhancing the system's high-dimensional beamforming capabilities and spatial electromagnetic control freedom.

[0003] While BD-RIS offers greater spatial control freedom, its fully connected circuit topology inevitably introduces non-convex hardware geometric constraints in joint waveform optimization design that are far more stringent than traditional constant modulus constraints. These constraints are complex symmetry (satisfying electromagnetic network reciprocity) and unitary constraints (satisfying passive and lossless characteristics). Traditional mathematical relaxation algorithms suffer from the following significant drawbacks when dealing with such high-dimensional matrix dual constraints:

[0004] Deep entanglement with multiple hard constraints is difficult to handle: the optimal solution must lie simultaneously in the intersection of the unitary manifold and the symmetric subspace. Finding the optimal solution on the intersection of two non-convex sets is mathematically challenging. Traditional gradient descent on manifolds is prone to causing iteration points to satisfy symmetry while simultaneously breaking orthogonality, and vice versa.

[0005] Reprojection computation is costly: Under unitary manifold constraints, conventional algorithms typically require highly complex operators such as singular value decomposition (SVD) to ensure that the matrix is ​​rolled back to the manifold surface in real time during the iterative update process. The computational cost is much higher than that of traditional RIS scalar projection, making it difficult to meet the real-time processing requirements of the ISAC system for highly dynamic channels.

[0006] Relaxation error leads to performance degradation: When traditional relaxation methods perform rank-1 approximation and Gaussian randomization recovery in complex matrix spaces, they introduce non-negligible truncation errors, resulting in a decrease in radar beam fitting accuracy and a loss of communication and data rate. Summary of the Invention

[0007] The purpose of this invention is to provide a syn-sensory integrated waveform control method based on a super-diagonal intelligent reflector, which solves the problems of deep entanglement of multiple hard constraints, high cost of reprojection calculation, and / or performance degradation caused by relaxation error in the existing technology.

[0008] This invention is achieved through the following technical solution:

[0009] A method for integrated sensing and waveform control based on a super-diagonal intelligent reflector surface includes:

[0010] Obtain the communication management mode of the communication system where the super-diagonal intelligent reflective surface is located; the communication management mode includes a flexible adjustment mode or a stable convergence mode.

[0011] Based on the communication management mode being the flexible adjustment mode, the Riemann manifold gradient descent method based on the Lagrange penalty function is used to obtain the integrated waveform control parameters of the syn-sensory system.

[0012] Based on the stable convergence mode of the communication management mode, the Riemann manifold gradient descent method based on Takagi decomposition is used to obtain the joint waveform control parameters of the syn-inductive integration.

[0013] The integrated sensing waveform control parameters are transmitted to the impedance matching network of the base station's transmitting radio frequency front end and BD-RIS to control the integrated sensing waveform of the integrated sensing signal and realize the transmission of the integrated sensing signal.

[0014] In one possible implementation, the Riemannian manifold gradient descent method based on the Lagrange penalty function is used to obtain the integrated sensing waveform control parameters, including:

[0015] The active transmit beam matrix of the base station, the reflection phase shift matrix of BD-RIS, and the power scaling factor are randomly generated in the communication system, and a copy of the reflection phase shift matrix is ​​obtained at the same time.

[0016] Based on the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the copy of the reflection phase shift matrix, and the power scaling factor, obtain a first joint optimization function with an equality constraint penalty between the reflection phase shift matrix and its copy;

[0017] Based on the first joint optimization function, the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, a copy of the reflection phase shift matrix, and the first Euclidean gradient corresponding to the power scaling factor are obtained respectively.

[0018] Based on the first Euclidean gradient, the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the copy of the reflection phase shift matrix, and the first Riemann gradient corresponding to the power scaling factor are obtained respectively.

[0019] The active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the copy of the reflection phase shift matrix, and the power scaling factor are used as variables. The search is performed along the direction of the first Riemann gradient descent until the first Riemann gradient converges. The updated variables are then projected back onto the manifold using the backtracking operator to obtain the updated active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the copy of the reflection phase shift matrix, and the power scaling factor.

[0020] If the number of iterations has reached the maximum number of iterations, then obtain the integrated waveform control parameters of the induction system based on the updated active transmit beam matrix and the updated BD-RIS reflection phase shift matrix; otherwise, obtain the residual data based on the updated BD-RIS reflection phase shift matrix and a copy of the updated reflection phase shift matrix, and adaptively adjust the first joint optimization function based on the residual data to obtain the adjusted first joint optimization function.

[0021] Based on the adjusted first joint optimization function, return to the step of obtaining the first Euclidean gradient and proceed to the next iteration.

[0022] In one possible implementation, based on the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, a copy of the reflection phase shift matrix, and a power scaling factor, a first joint optimization function with an equality constraint penalty between the reflection phase shift matrix and its copy is obtained as follows:

[0023] ;

[0024] ;

[0025] ;

[0026] ;

[0027] = ;

[0028] = ;

[0029] ;

[0030] in, Denotes the first joint optimization function. Indicates the weighting coefficient. This represents the radar beam fitting error. This indicates interference error in multi-user communication. Indicates the penalty factor; The reflection phase shift matrix is ​​represented by an off-diagonal matrix, and the response of BD-RIS is characterized by this matrix. as well as ; Let A represent the identity matrix of order A, where A represents the number of elements in the BD-RIS matrix. This represents a copy of the reflection phase shift matrix. The dual variable representing the reflection phase shift matrix, Indicates the sampling point number in the coverage area. Indicates the total number of sampling points. Indicates the directional weighting coefficient. Indicates will Expand and stack them column-wise into a vector. Indicates the active transmit beam matrix, superscript This indicates the conjugate transpose. This represents the radar information matrix characterizing the spatial power distribution. Indicates the power scaling factor. This represents the expected radiated power in the direction of the k-th sampling point. Denotes an L-order identity matrix. Indicates the Kronecker product, superscript The superscript T indicates conjugation, and the superscript T indicates transpose. The guide vector represents a uniform linear array. Represents the natural constant. Represents the imaginary number symbol, This represents the k-th angle. Indicates the number of antennas. This represents the extended equivalent synthesized channel matrix. Indicates will Expand and stack them into a vector by columns. Represents the desired communication symbol matrix. This represents the equivalent composite channel, where N represents the number of users and L represents the total number of discrete-time snapshots. Represents the field of complex numbers. This refers to the direct link channel from the base station to the user. This represents the reflection link channel between the user and BD-RIS. This indicates the link channel between BD-RIS and the base station.

[0031] In one possible implementation, the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, a copy of the reflection phase shift matrix, and the first Euclidean gradient corresponding to the power scaling factor are obtained according to the first joint optimization function, including:

[0032] ;

[0033] ;

[0034] ;

[0035] ;

[0036] in, This represents the first Euclidean gradient of the active transmit beam matrix. This represents the first Euclidean gradient of the reflection phase shift matrix of BD-RIS. Let represent the first Euclidean gradient of the copy of the reflection phase shift matrix. This represents the first Euclidean gradient of the power scaling factor. This represents the detection power at the k-th angle. Indicates the discrete-time snapshot number. This represents the total number of discrete-time snapshots. This indicates the BD-RIS to user reflection link channel. This indicates the reflection link channel from the base station to BD-RIS. Represents the active transmit beam matrix The column components, The desired communication symbol matrix S represents the first... Column components.

[0037] In one possible implementation, based on the first Euclidean gradient, the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, a copy of the reflection phase shift matrix, and the first Riemann gradient corresponding to the power scaling factor are obtained, including:

[0038] ;

[0039] ;

[0040] ;

[0041] = ;

[0042] in, This represents the first Riemann gradient of the active transmit beam matrix. This represents the first Riemann gradient of the reflection phase shift matrix of BD-RIS. This represents the first Riemann gradient of a copy of the reflection phase shift matrix. Indicates taking the real part, It represents the Hadamardi (or Hadama) stack.

[0043] In one possible implementation, the active transmit beam matrix, the BD-RIS reflection phase shift matrix, a copy of the reflection phase shift matrix, and the power scaling factor are used as variables. The search proceeds along the direction of the first Riemann gradient descent until the first Riemann gradient converges. The updated variables are then projected back onto the manifold using a backtracking operator to obtain the updated active transmit beam matrix, the BD-RIS reflection phase shift matrix, a copy of the reflection phase shift matrix, and the power scaling factor, including:

[0044] The active transmit beam matrix, the reflection phase shift matrix of BD-RIS, a copy of the reflection phase shift matrix, and the power scaling factor are used as variables to search along the direction of the first Riemann gradient descent.

[0045] The search is repeated until the first Riemann gradient converges. The updated variables are then projected back onto the manifold using the backtracking operator to obtain the updated active transmit beam matrix, the BD-RIS reflection phase shift matrix, a copy of the reflection phase shift matrix, and the power scaling factor:

[0046] ;

[0047] ;

[0048] ;

[0049] ;

[0050] in, This represents the active transmit beam matrix after being updated during the h-th search along the direction of the first Riemann gradient descent. This represents the active transmission beam matrix updated during the h-1th search along the direction of the first Riemann gradient descent. This indicates that the columns are stacked to form a vector. This indicates the step size selected according to the Armijo criterion. Let f(x) represent the reflection phase shift matrix of BD-RIS after the h-th search along the direction of the first Riemann gradient descent. This represents the reflection phase shift matrix of the BD-RIS after being updated during the (h-1)th search along the direction of the first Riemann gradient descent. This represents a copy of the reflection phase shift matrix after the update during the h-th search along the direction of the first Riemann gradient descent. This represents a copy of the reflection phase shift matrix after the update during the (h-1)th search along the direction of the first Riemann gradient descent. This indicates a heavy traction operation, and , This represents the variable matrix in the heavy traction operation. ; To pass through the variable matrix The first singular value matrix obtained by performing singular value decomposition. To pass through the variable matrix The second singular value matrix obtained by performing singular value decomposition. Represents a singular value matrix. This represents the power scaling factor updated during the h-th search along the direction of the first Riemann gradient descent. This represents the updated power scaling factor during the (h-1)th search along the direction of the first Riemann gradient descent.

[0051] In one possible implementation, residual data is obtained based on the updated BD-RIS reflection phase shift matrix and a copy of the updated reflection phase shift matrix, and the first joint optimization function is adaptively adjusted based on the residual data to obtain the adjusted first joint optimization function, including:

[0052] Based on the reflection phase shift matrix of the BD-RIS and a copy of the reflection phase shift matrix, the residual data is obtained as follows:

[0053] ;

[0054] in, Let represent the reflection phase shift matrix of BD-RIS after the update in the t-th iteration. This represents a copy of the reflection phase shift matrix after the update during the t-th iteration. Denotes the Frobenius norm. This represents the residual data during the t-th iteration.

[0055] If the residual data is greater than or equal to a preset residual threshold, the penalty factor in the first joint optimization function is updated as follows:

[0056] ;

[0057] in, This represents the penalty factor in the t-th iteration. This represents the penalty factor in the (t+1)th iteration, i.e., the penalty factor after the update; This represents the attenuation coefficient between (0,1);

[0058] If the residual data is less than a preset residual threshold, then the dual variable in the first joint optimization function is updated:

[0059] ;

[0060] in, Let represent the dual variable in the t-th iteration. Let represent the dual variable in the (t+1)th iteration, i.e., the dual variable after the update;

[0061] The first joint optimization function with the updated penalty factor or dual variable is used as the adjusted first joint optimization function.

[0062] In one possible implementation, the joint waveform control parameters for synesthesia integration are obtained using a Riemannian manifold gradient descent method based on Takagi decomposition, including:

[0063] Randomly generate the active transmit beam matrix of the base station, the reflection phase shift matrix of BD-RIS, and the power scaling factor in the communication system;

[0064] The reflection phase shift matrix of the BD-RIS is decomposed into a product of unitary matrices according to the Takagi decomposition principle: ;in, This represents the reflection phase shift matrix of BD-RIS. This represents an auxiliary matrix, with the superscript T indicating transpose.

[0065] Based on the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, and the power scaling factor, the second joint optimization function for fusing the radar beam fitting weighted mean square error and the multi-user communication interference error is obtained as follows: ; ; = ;

[0066] The second Euclidean gradients corresponding to the active transmit beam matrix, power scaling factor, and auxiliary matrix are obtained according to the second joint optimization function, and the second Riemann gradients corresponding to the active transmit beam matrix, auxiliary matrix, and power scaling factor are obtained according to the second Euclidean gradients.

[0067] The search is performed along the second Riemann gradient descent direction to obtain the active transmit beam matrix and auxiliary matrix after the search, and the integrated waveform control parameters of the induction system are obtained based on the active transmit beam matrix and auxiliary matrix after the search.

[0068] In one possible implementation, the second Euclidean gradients corresponding to the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the power scaling factor, and the auxiliary matrix are obtained according to the second joint optimization function, and the second Riemann gradients corresponding to the active transmit beam matrix, the auxiliary matrix, and the power scaling factor are obtained according to the second Euclidean gradients, including:

[0069] Based on the second joint optimization function, the second Euclidean gradients corresponding to the active transmit beam matrix, power scaling factor, and auxiliary matrix are obtained as follows:

[0070] ;

[0071] ;

[0072] ;

[0073] ;

[0074] in, This represents the second Euclidean gradient of the active transmission beam matrix. This represents the second Euclidean gradient of the reflection phase shift matrix of BD-RIS. This represents the second Euclidean gradient of the power scaling factor. This represents the second Euclidean gradient of the auxiliary matrix;

[0075] Based on the second Euclidean gradient, the second Riemann gradients corresponding to the active transmit beam matrix, auxiliary matrix, and power scaling factor are obtained as follows:

[0076] ;

[0077] ;

[0078] = ;

[0079] in, This represents the second Riemann gradient of the auxiliary matrix. This represents the second Riemann gradient of the active transmission beam matrix. This represents the second Riemann gradient of the power scaling factor. This represents the symmetry operator.

[0080] In one possible implementation, a search is performed along the second Riemann gradient descent direction to obtain the searched active transmit beam matrix and auxiliary matrix, and the integrated sensing waveform control parameters are obtained based on the searched active transmit beam matrix and auxiliary matrix, including:

[0081] The search proceeds along the descent direction of the second Riemann gradient until the second Riemann gradient converges. The updated variables are then projected back onto the manifold using the backtracking operator, yielding the active transmit beam matrix and auxiliary matrix after the search:

[0082] ;

[0083] ;

[0084] ;

[0085] in, Indicates the first The active transmission beam matrix during the search process along the second Riemann gradient descent direction, i.e., the active transmission beam matrix after the search. Indicates the current search count. Indicates the first The active transmission beam matrix during the search process along the second Riemann gradient descent direction for the -1st time. Let represent the auxiliary matrix used in the t-th search along the second Riemann gradient descent direction. Indicates the first -1 auxiliary matrix during the search process along the second Riemann gradient descent direction. This represents the extraction of orthogonal matrix factors in the QR decomposition of the matrix. Represents the updated matrix , This represents the orthogonal matrix obtained after performing QR decomposition on V;

[0086] The optimal active transmission beam matrix is ​​obtained based on the searched active transmission beam matrix. The optimal reflection phase shift matrix is ​​obtained based on the auxiliary matrix after the search. The optimal active transmission beam matrix and the optimal reflection phase shift matrix are used together as the integrated waveform control parameters for the sensing system; wherein, This represents the auxiliary matrix after the search, i.e., the optimal auxiliary matrix.

[0087] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0088] This invention discloses a joint waveform control method for integrated sensing and communication based on a super-diagonal intelligent reflector. The method first obtains the agile adjustment or stable convergence mode of the communication system. For different modes, it employs the Riemann manifold gradient descent method based on the Lagrange penalty function and Takagi decomposition to solve for the optimal base station transmit beam matrix and the optimal BD-RIS reflection phase shift matrix. These optimal base station transmit beam matrix and optimal BD-RIS reflection phase shift matrix are then transmitted to the base station RF front-end and the BD-RIS impedance matching network, achieving precise waveform control of the integrated sensing and communication signal. This effectively improves communication and sensing performance, avoids the numerical oscillations and hyperparameter sensitivity caused by the double-layer loop of the penalty function method, simplifies the algorithm solution structure, and effectively approximates the theoretical performance upper bound of the system while strictly satisfying physical constraints. It is suitable for complex and dynamic integrated sensing and communication scenarios. Attached Figure Description

[0089] To more clearly illustrate the technical solutions of the exemplary embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly described below. It should be understood that the following drawings only show some embodiments of the present invention and should not be considered as a limitation of the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort. In the drawings:

[0090] Figure 1 A flowchart of a syn-sensory integrated waveform control method based on a super-diagonal smart reflector provided in this application embodiment;

[0091] Figure 2 A schematic diagram illustrating the application scenarios provided in the embodiments of this application;

[0092] Figure 3 A schematic diagram of the Riemannian manifold gradient descent method based on the Lagrange penalty function provided in this application embodiment;

[0093] Figure 4 This is a schematic diagram of the Riemannian manifold gradient descent method based on Takagi decomposition provided in an embodiment of this application. Detailed Implementation

[0094] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the embodiments and accompanying drawings. The illustrative embodiments and descriptions of the present invention are only used to explain the present invention and are not intended to limit the present invention.

[0095] like Figure 1 As shown, this application provides a method for integrated sensing and waveform control based on a super-diagonal smart reflector, comprising:

[0096] S101. Obtain the communication management mode of the communication system where the super-diagonal intelligent reflective surface is located; the communication management mode includes a flexible adjustment mode or a stable convergence mode.

[0097] S102. Based on the communication management mode being the flexible adjustment mode, the Riemann manifold gradient descent method based on the Lagrange penalty function is used to obtain the integrated waveform control parameters of the syn-sensory system.

[0098] S103. Based on the stable convergence mode of the communication management mode, the Riemann manifold gradient descent method based on Takagi decomposition is used to obtain the joint waveform control parameters of the syn-inductive integration.

[0099] S104. The integrated sensing waveform control parameters are transmitted to the impedance matching network of the base station transmitting radio frequency front end and BD-RIS to control the integrated sensing waveform of the integrated sensing signal and realize the transmission of the integrated sensing signal.

[0100] This invention proposes a soft-constraint solution mechanism based on the Lagrange penalty function and a hard-constraint manifold equivalent reconstruction mechanism based on Takagi decomposition, aiming to avoid the approximation error introduced by traditional relaxation algorithms. Under the premise of satisfying the BD-RIS physical hardware constraints, it achieves synergy between radar beam fitting and multi-user communication interference suppression, thereby improving the system's spatial control freedom and overall performance.

[0101] To facilitate understanding of the technical solutions described in the embodiments of this application by those skilled in the art, the application scenarios of the technical solutions described in the embodiments of this application are first introduced. For example... Figure 2 As shown, the BD-RIS-assisted downlink sensing and communication integrated system consists of a multi-antenna base station, a super-diagonal smart reflector, multiple single-antenna communication users, and a radar target. The base station needs to simultaneously transmit data to multiple communication users and use radar beams to detect the target area.

[0102] Assuming the base station is equipped with Root transmitting antenna, in discrete time snapshot Internal signal transmission This signal is also used to send... The system transmits information to multiple communication users and performs radar sensing of the airspace of multiple targets. The system introduces a... The BD-RIS of each array element is used to enhance channel modulation capabilities. This is a direct link channel from the base station to the user. This is the link channel between BD-RIS and the base station. This is the reflection link channel between the user and the BD-RIS. Unlike traditional diagonal RIS, the response of the BD-RIS consists of an off-diagonal matrix. Indicated. Based on the circuit topology and passive characteristics, The complex symmetric constraint and the unitary constraint must be satisfied, that is:

[0103] ;

[0104] ;

[0105] In the In a snapshot, the user's received signal vector is represented as follows:

[0106] ;

[0107] in, This is additive white Gaussian noise. (Note: The original text contains some formatting errors and inconsistencies. A more For the desired communication symbol matrix, This is an equivalent composite channel. Communication performance is measured by minimizing multi-user interference:

[0108] ;

[0109] Among them, let = For the extended equivalent synthesized channel matrix, Expand and stack them column-wise to form a vector s. Expand and stack the columns to form a vector x. The above formula can be expressed as:

[0110] ;

[0111] Radar in a specific discrete spatial direction The power of the composite pattern at the point is expressed as

[0112] ;

[0113] in, A radar information matrix characterizing the spatial power distribution properties. for Expanded and stacked column-wise into a vector, k is the sampling point index of the coverage space, and the guide vector of the uniform linear array. The expression is as follows:

[0114] ;

[0115] Introducing the desired beam pattern power distribution vector Directional weighting coefficient and power scaling factor By minimizing the weighted deviation between the actual detection power and the desired template, a sensing design objective function is constructed:

[0116] ;

[0117] The system aims to jointly optimize the base station's active transmit beam matrix by minimizing multi-user communication interference and radar beam fitting error. Passive reflection phase shift matrix of BD-RIS The final joint optimization model is obtained as follows:

[0118] ;

[0119] ;

[0120] For the above optimization model, this invention proposes two progressive solution schemes:

[0121] The Riemannian gradient descent method based on the Lagrange penalty function: This scheme transforms the difficult-to-handle dual geometric constraints (the intersection of symmetric and unitary constraints) into a sequence of soft-constrained subproblems on the unitary manifold by introducing dual variables and a quadratic penalty term. The penalty factor is dynamically updated in the outer loop, while the Riemannian gradient descent method is used to find the optimal solution on the unitary manifold in the inner loop.

[0122] Riemannian Gradient Descent Method Based on Takagi Decomposition: To overcome the limitations of the penalty function method in Scheme 1, which is extremely sensitive to hyperparameters and suffers from numerical fluctuations, this scheme utilizes the Takagi decomposition property of complex symmetric unitary matrices to parameterize the original matrix as follows: Through this deep geometric reconstruction, the complex dual constraints are losslessly and equivalently mapped to orthogonal constraints on the standard complex Stiefel manifold. Stable and high-fidelity single-layer joint optimization is achieved directly on the product manifold space.

[0123] In some possible implementations, based on the two progressive solution schemes proposed above, the inductive integrated waveform control method based on a super-diagonal intelligent reflective surface proposed in this application will be further explained.

[0124] At the start of the calculation, the number of base station transmit antennas can be obtained. Number of communication users , BD-RIS array element number and the number of snapshots Obtain the channel matrices from base station to user, base station to BD-RIS, and BD-RIS to user, as well as the expected radar detection beam pattern and communication symbol matrix. Construct a joint sensing and communication optimization objective function. Based on the base station's active transmit beam matrix... With BD-RIS reflection matrix We construct a weighted joint cost function that includes the mean square error of radar beam fitting and the multi-user interference error of communication.

[0125] Then, a solution scheme is selected. If the system tolerates a certain parameter tuning overhead to gain flexibility, the soft constraint mode is executed; if the system requires absolute numerical stability and low-latency convergence, the hard constraint mode is executed.

[0126] For example, obtaining the communication management mode of the communication system where the super-diagonal smart reflector is located includes:

[0127] The communication management mode preset by the user or directly input by the user through human-computer interaction is obtained from the designated data storage area of ​​the communication system where the super-diagonal intelligent reflector is located. The communication management mode includes a flexible adjustment mode or a stable convergence mode. The flexible adjustment mode means that the communication system allows a certain amount of parameter tuning overhead. The stable convergence mode means that the communication system allows absolute numerical stability and low-latency convergence. For example, if the system tolerates the existence of residual values ​​from the first method and they are less than a preset threshold, the first method based on the penalty function can be used. If the system requires the processing delay to be lower than the channel coherence time or requires strict adherence to physical constraints, i.e., the channel information remains unchanged during this time, the second decomposition method is used. The flowchart of the Riemannian manifold gradient descent method based on the Lagrange penalty function is as follows: Figure 3 As shown; the flowchart of the Riemannian manifold gradient descent method based on Takagi decomposition is as follows: Figure 4 As shown.

[0128] In one possible implementation, the Riemannian manifold gradient descent method based on the Lagrange penalty function is used to obtain the integrated sensing waveform control parameters, including:

[0129] The active transmit beam matrix of the base station in the communication system, the reflection phase shift matrix of BD-RIS, and the power scaling factor are randomly generated, and a copy of the reflection phase shift matrix is ​​obtained simultaneously. For example, lumped parameters can be randomly generated based on constraints. For instance, the waveform is generated as a matrix that meets constant modulus constraints, the reflection phase shift matrix of BD-RIS is generated as a matrix that meets complex symmetry and unitary constraints, and the power scaling factor can be generated as 1.

[0130] Based on the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the copy of the reflection phase shift matrix, and the power scaling factor, obtain a first joint optimization function with an equality constraint penalty between the reflection phase shift matrix and its copy;

[0131] Based on the first joint optimization function, the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, a copy of the reflection phase shift matrix, and the first Euclidean gradient corresponding to the power scaling factor are obtained respectively.

[0132] Based on the first Euclidean gradient, the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the copy of the reflection phase shift matrix, and the first Riemann gradient corresponding to the power scaling factor are obtained respectively.

[0133] The active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the copy of the reflection phase shift matrix, and the power scaling factor are used as variables. The search is performed along the direction of the first Riemann gradient descent until the first Riemann gradient converges. The updated variables are then projected back onto the manifold using the backtracking operator to obtain the updated active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the copy of the reflection phase shift matrix, and the power scaling factor.

[0134] If the number of iterations has reached the maximum number of iterations, then obtain the integrated waveform control parameters of the induction system based on the updated active transmit beam matrix and the updated BD-RIS reflection phase shift matrix; otherwise, obtain the residual data based on the updated BD-RIS reflection phase shift matrix and a copy of the updated reflection phase shift matrix, and adaptively adjust the first joint optimization function based on the residual data to obtain the adjusted first joint optimization function.

[0135] Based on the adjusted first joint optimization function, return to the step of obtaining the first Euclidean gradient and proceed to the next iteration.

[0136] In one possible implementation, based on the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, a copy of the reflection phase shift matrix, and a power scaling factor, a first joint optimization function with an equality constraint penalty between the reflection phase shift matrix and its copy is obtained as follows:

[0137] ;

[0138] ;

[0139] ;

[0140] ;

[0141] = ;

[0142] = ;

[0143] ;

[0144] in, Denotes the first joint optimization function. Indicates the weighting coefficient. This represents the radar beam fitting error. This indicates interference error in multi-user communication. Indicates the penalty factor; The reflection phase shift matrix is ​​represented by an off-diagonal matrix, and the response of BD-RIS is characterized by this matrix. as well as ; Let A represent the identity matrix of order A, where A represents the number of elements in the BD-RIS matrix. This represents a copy of the reflection phase shift matrix. The dual variable representing the reflection phase shift matrix, Indicates the sampling point number in the coverage area. Indicates the total number of sampling points. Indicates the directional weighting coefficient. Indicates will Expand and stack them into a vector by columns. Indicates the active transmit beam matrix, superscript This indicates the conjugate transpose. This represents the radar information matrix characterizing the spatial power distribution. Indicates the power scaling factor. This represents the expected radiated power in the direction of the k-th sampling point. Denotes an L-order identity matrix. Indicates the Kronecker product, superscript The superscript T indicates conjugation, and the superscript T indicates transpose. The guide vector represents a uniform linear array. Represents the natural constant. Represents the imaginary number symbol; This represents the k-th angle, which divides the space to be probed into K angles, here the k-th angle; Indicates the number of antennas. This represents the extended equivalent synthesized channel matrix. Indicates will Expand and stack them into a vector by columns. Represents the desired communication symbol matrix. This represents the equivalent composite channel, where N represents the number of users and L represents the total number of discrete-time snapshots. Represents the field of complex numbers. This refers to the direct link channel from the base station to the user. This represents the reflection link channel between the user and BD-RIS. This indicates the link channel between BD-RIS and the base station.

[0145] The method for obtaining the first joint optimization function mentioned above may include:

[0146] Introducing the reflection matrix copy Apply unitary constraints to The complex symmetric constraint is preserved in Through this variable splitting strategy, the complex constraints of a single variable are transformed into... and The equality constraints between them transform the original joint optimization model into:

[0147] ;

[0148] ;

[0149] Introducing the reflection matrix dual variables With penalty factor .Will and The equality constraints between the equations are penalized and incorporated into the weighted joint cost function, transforming the joint optimization model into:

[0150]

[0151]

[0152] Then in , , and The accumulation shape The above joint optimization model is an unconstrained model, that is...

[0153]

[0154] In one possible implementation, the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, a copy of the reflection phase shift matrix, and the first Euclidean gradient corresponding to the power scaling factor are obtained according to the first joint optimization function, including:

[0155] ;

[0156] ;

[0157] ;

[0158] ;

[0159] in, This represents the first Euclidean gradient of the active transmit beam matrix. This represents the first Euclidean gradient of the reflection phase shift matrix of BD-RIS. Let represent the first Euclidean gradient of the copy of the reflection phase shift matrix. This represents the first Euclidean gradient of the power scaling factor. This represents the detection power at the k-th angle. Indicates the discrete-time snapshot number. This represents the total number of discrete-time snapshots. This indicates the BD-RIS to user reflection link channel. This indicates the reflection link channel from the base station to BD-RIS. Represents the active transmit beam matrix The column components, The desired communication symbol matrix S represents the first... Column components.

[0160] In one possible implementation, based on the first Euclidean gradient, the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, a copy of the reflection phase shift matrix, and the first Riemann gradient corresponding to the power scaling factor are obtained, including:

[0161] ;

[0162] ;

[0163] ;

[0164] = ;

[0165] in, This represents the first Riemann gradient of the active transmit beam matrix. This represents the first Riemann gradient of the reflection phase shift matrix of BD-RIS. This represents the first Riemann gradient of a copy of the reflection phase shift matrix. Indicates taking the real part, It represents the Hadamardi (or Hadama) stack.

[0166] In one possible implementation, the active transmit beam matrix, the BD-RIS reflection phase shift matrix, a copy of the reflection phase shift matrix, and the power scaling factor are used as variables. The search proceeds along the direction of the first Riemann gradient descent until the first Riemann gradient converges. The updated variables are then projected back onto the manifold using a backtracking operator to obtain the updated active transmit beam matrix, the BD-RIS reflection phase shift matrix, a copy of the reflection phase shift matrix, and the power scaling factor, including:

[0167] The active transmit beam matrix, the reflection phase shift matrix of BD-RIS, a copy of the reflection phase shift matrix, and the power scaling factor are used as variables to search along the direction of the first Riemann gradient descent.

[0168] The search is repeated until the first Riemann gradient converges. The updated variables are then projected back onto the manifold using the backtracking operator to obtain the updated active transmit beam matrix, the BD-RIS reflection phase shift matrix, a copy of the reflection phase shift matrix, and the power scaling factor:

[0169] ;

[0170] ;

[0171] ;

[0172] ;

[0173] in, This represents the active transmit beam matrix after being updated during the h-th search along the direction of the first Riemann gradient descent. This represents the active transmission beam matrix updated during the h-1th search along the direction of the first Riemann gradient descent. This indicates that the columns are stacked to form a vector. This indicates the step size selected according to the Armijo criterion. Let f(x) represent the reflection phase shift matrix of BD-RIS after the h-th search along the direction of the first Riemann gradient descent. This represents the reflection phase shift matrix of the BD-RIS after being updated during the (h-1)th search along the direction of the first Riemann gradient descent. This represents a copy of the reflection phase shift matrix after the update during the h-th search along the direction of the first Riemann gradient descent. This represents a copy of the reflection phase shift matrix after the update during the (h-1)th search along the direction of the first Riemann gradient descent. This indicates a heavy traction operation, and , This represents the variable matrix in the heavy traction operation. ; To pass through the variable matrix The first singular value matrix obtained by performing singular value decomposition. To pass through the variable matrix The second singular value matrix obtained by performing singular value decomposition. Represents a singular value matrix. This represents the power scaling factor updated during the h-th search along the direction of the first Riemann gradient descent. This represents the updated power scaling factor during the (h-1)th search along the direction of the first Riemann gradient descent.

[0174] In one possible implementation, residual data is obtained based on the updated BD-RIS reflection phase shift matrix and a copy of the updated reflection phase shift matrix, and the first joint optimization function is adaptively adjusted based on the residual data to obtain the adjusted first joint optimization function, including:

[0175] Based on the reflection phase shift matrix of the BD-RIS and a copy of the reflection phase shift matrix, the residual data is obtained as follows:

[0176] ;

[0177] in, Let represent the reflection phase shift matrix of BD-RIS after the update in the t-th iteration. This represents a copy of the reflection phase shift matrix after the update during the t-th iteration. Denotes the Frobenius norm. This represents the residual data during the t-th iteration.

[0178] If the residual data is greater than or equal to a preset residual threshold, the penalty factor in the first joint optimization function is updated as follows:

[0179] ;

[0180] in, This represents the penalty factor in the t-th iteration. This represents the penalty factor in the (t+1)th iteration, i.e., the penalty factor after the update; This represents the attenuation coefficient between (0,1);

[0181] If the residual data is less than a preset residual threshold, then the dual variable in the first joint optimization function is updated:

[0182] ;

[0183] in, Let represent the dual variable in the t-th iteration. Let represent the dual variable in the (t+1)th iteration, i.e., the dual variable after the update;

[0184] The first joint optimization function with the updated penalty factor or dual variable is used as the adjusted first joint optimization function.

[0185] For example, the penalty factor and dual variable can be dynamically updated based on the degree of constraint violation, and the Frobenius norm of the difference between the reflection matrix and the replica can be taken as the residual, i.e. .like If the penalty term is large or decreases slowly, the algorithm introduces a decay coefficient to increase the weight of the penalty term in the objective function, forcing the optimization variables to quickly approach the consistent feasible region, i.e., updating the penalty factor. ( (This is the attenuation coefficient). Conversely, if the residuals have decreased to within the preset threshold or show a robust decreasing trend, the penalty factor remains stable, and the current residuals are used to update the dual variable for refined compensation. This improves the accuracy of constraint satisfaction while avoiding excessively large penalty terms from obscuring the original synesthetic optimization objective, until the convergence condition is finally met, i.e., the dual variable is updated. .

[0186] In one possible implementation, the joint waveform control parameters for synesthesia integration are obtained using a Riemannian manifold gradient descent method based on Takagi decomposition, including:

[0187] Randomly generate the active transmit beam matrix of the base station, the reflection phase shift matrix of BD-RIS, and the power scaling factor in the communication system;

[0188] The reflection phase shift matrix of the BD-RIS is decomposed into a product of unitary matrices according to the Takagi decomposition principle: ;in, This represents the reflection phase shift matrix of BD-RIS. This represents an auxiliary matrix, with the superscript T indicating transpose.

[0189] Based on the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, and the power scaling factor, the second joint optimization function for fusing the radar beam fitting weighted mean square error and the multi-user communication interference error is obtained as follows: ; ; = ;

[0190] The second Euclidean gradients corresponding to the active transmit beam matrix, power scaling factor, and auxiliary matrix are obtained according to the second joint optimization function, and the second Riemann gradients corresponding to the active transmit beam matrix, auxiliary matrix, and power scaling factor are obtained according to the second Euclidean gradients.

[0191] The search is performed along the second Riemann gradient descent direction to obtain the active transmit beam matrix and auxiliary matrix after the search, and the integrated waveform control parameters of the induction system are obtained based on the active transmit beam matrix and auxiliary matrix after the search.

[0192] In one possible implementation, the method for obtaining the second joint optimization function may include: using the reflection matrix According to the principles of Takagi decomposition, it can be decomposed into a product of unitary matrices, that is: Auxiliary matrix It belongs to a complex Stiefel manifold, therefore in , and The accumulation shape The original joint optimization model can be expressed as: .

[0193] In one possible implementation, the second Euclidean gradients corresponding to the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the power scaling factor, and the auxiliary matrix are obtained according to the second joint optimization function, and the second Riemann gradients corresponding to the active transmit beam matrix, the auxiliary matrix, and the power scaling factor are obtained according to the second Euclidean gradients, including:

[0194] Based on the second joint optimization function, the second Euclidean gradients corresponding to the active transmit beam matrix, power scaling factor, and auxiliary matrix are obtained as follows:

[0195] ;

[0196] ;

[0197] ;

[0198] ;

[0199] in, This represents the second Euclidean gradient of the active transmission beam matrix. This represents the second Euclidean gradient of the reflection phase shift matrix of BD-RIS. This represents the second Euclidean gradient of the power scaling factor. The auxiliary matrix represents the second Euclidean gradient of the auxiliary matrix. The Euclidean gradient can be obtained by applying... The Euclidean gradient is calculated using the chain rule;

[0200] Then, the Euclidean gradient obtained in the previous step can be projected. Since each variable corresponds to different geometric manifold constraints, the corresponding tangent space projection operator needs to be applied separately to orthogonally project the Euclidean gradient onto the tangent space of its respective manifold to obtain the Riemann gradient for iterative updates. The auxiliary matrix... The Euclidean gradient is orthogonally projected onto the tangent space of the Stiefel manifold.

[0201] Based on the second Euclidean gradient, the second Riemann gradients corresponding to the active transmit beam matrix, auxiliary matrix, and power scaling factor are obtained as follows:

[0202] ;

[0203] ;

[0204] = ;

[0205] in, This represents the second Riemann gradient of the auxiliary matrix. This represents the second Riemann gradient of the active transmission beam matrix. This represents the second Riemann gradient of the power scaling factor. This represents the symmetry operator. For symmetrization operators, The variable representing the symmetry operator, this projection guarantees that the direction of iteration on the complex Stiefel manifold maintains the absolute orthogonality of the matrix column vectors.

[0206] In one possible implementation, a search is performed along the second Riemann gradient descent direction to obtain the searched active transmit beam matrix and auxiliary matrix, and the integrated sensing waveform control parameters are obtained based on the searched active transmit beam matrix and auxiliary matrix, including:

[0207] The search proceeds along the descent direction of the second Riemann gradient until the second Riemann gradient converges. The updated variables are then projected back onto the manifold using the backtracking operator, yielding the active transmit beam matrix and auxiliary matrix after the search:

[0208] ;

[0209] ;

[0210] ;

[0211] in, Indicates the first The active transmission beam matrix during the search process along the second Riemann gradient descent direction, i.e., the active transmission beam matrix after the search. Indicates the current search count. Indicates the first The active transmission beam matrix during the search process along the second Riemann gradient descent direction for the -1st time. Let represent the auxiliary matrix used in the t-th search along the second Riemann gradient descent direction. Indicates the first -1 auxiliary matrix during the search process along the second Riemann gradient descent direction. This represents the extraction of orthogonal matrix factors in the QR decomposition of the matrix. Represents the updated matrix , This represents the orthogonal matrix obtained after performing QR decomposition on V.

[0212] The optimal active transmission beam matrix is ​​obtained based on the searched active transmission beam matrix. The optimal reflection phase shift matrix is ​​obtained based on the auxiliary matrix after the search. The optimal active transmission beam matrix and the optimal reflection phase shift matrix are used together as the integrated waveform control parameters for the sensing system; wherein, This represents the auxiliary matrix after the search, i.e., the optimal auxiliary matrix. For example, it can be obtained from the convergence of the optimization. and (or Configure the base station's transmitting radio frequency front-end and the impedance matching network of BD-RIS to complete the transmission of integrated sensing signals.

[0213] This application breaks through the physical limitations of traditional diagonal RIS and proposes two effective manifold optimization paradigms to address the dual non-convex constraints of complex symmetry and unitary matrices introduced in waveform design of fully connected BD-RIS architectures. The problem of finding the intersection of these dual geometric constraints is decoupled, and the symmetry constraints are softened through a dynamic quadratic penalty term, providing a fundamental solution framework for complex high-dimensional non-convex feasible regions. Takagi decomposition is used to equivalently map the highly coupled complex symmetry and unitary constraints to a standard complex Stiefel manifold. This method avoids the numerical oscillations and hyperparameter sensitivity caused by the double-layer loop of the penalty function method, simplifies the algorithm's solution structure, and effectively approximates the theoretical performance upper bound of the system while strictly satisfying the physical constraints.

[0214] Based on the above technical solution, the embodiments of this application provide further examples as follows.

[0215] Taking a typical BD-RIS-assisted integrated sensing network service scenario as an example, assuming the base station is equipped with M transmitting antennas, and a BD-RIS containing A array elements serves N single-antenna users, performing radar target perception at several specific azimuth angles within space, while setting the time snapshot number to L. The specific implementation steps are as follows:

[0216] Step 1, Parameter Initialization and Information Acquisition:

[0217] Upon system startup, the system first acquires forward instantaneous channel state information from the base station to the user, from the base station to BD-RIS, and from BD-RIS to the user through channel estimation. The base station then generates the user's desired communication symbol matrix. And set the desired beam pattern power distribution vector for radar sensing. (e.g., a flat-top beam with a high level in the target area and a low level in the sidelobe area).

[0218] Step 2: Construct a joint optimization objective:

[0219] The base station processing unit constructs a joint cost function based on the acquired channel matrix and system parameters, with the radar beam fitting weighted mean square error and multi-user interference in communication as its core components. Weighting factors are assigned to the priorities of "communication" and "sensing" according to the current scenario (e.g., setting...). (Balancing the performance of both)

[0220] Step 3, Select a solution method:

[0221] Considering the rapid time-varying characteristics of high-frequency channels, the system requires the optimization algorithm to possess extremely high convergence stability and low computational latency. Therefore, the system scheduling center chose to execute the Riemannian manifold gradient descent method based on Takagi decomposition to decouple the dual hardware constraints of BD-RIS.

[0222] Step 4, perform manifold iterative optimization:

[0223] Geometric Reconstruction: The algorithm converts the off-diagonal reflection matrix of BD-RIS Parameterization is represented as At this time, the auxiliary matrix It automatically satisfies complex symmetric constraints, and its optimization space is transformed into a standard complex Stiefel manifold.

[0224] Gradient calculation and projection: In the product manifold space, the algorithm first calculates the joint cost function for the base station transmit beam matrix. Auxiliary matrix and power scaling factor The Euclidean gradient is obtained; then, the manifold tangent space projection operator is applied to orthogonally project the Euclidean gradient onto the tangent space of the corresponding complex circular manifold and the Stiefel manifold, thus obtaining the Riemann gradient that guides the correct descent.

[0225] Step size search and backtracking: A line search is performed along the negative direction of the Riemann gradient, and the optimal step size for the current iteration is automatically determined using the Armijo criterion; after updating the variables, the auxiliary matrix deviating from the surface is backtracked using the backtracking operator. Reprojecting back onto the Stiefel manifold, The projection restores the circular manifold.

[0226] Iterative convergence: Repeat the gradient projection and backtracking steps until the norm of the Riemann gradient is lower than the preset convergence threshold, at which point the algorithm stops.

[0227] Step 5, Output the solution and system deployment:

[0228] The base station outputs the optimal transmit waveform matrix based on the algorithm. Configure the phase and amplitude weights of the RF front-end and begin transmitting the integrated sensing signal. Simultaneously, based on the converged solution... The optimal reflection matrix is ​​obtained through reconstruction. The matrix instruction is then sent to the BD-RIS controller, which dynamically configures the interconnect impedance network between array elements.

[0229] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for integrated sensing and waveform control based on a super-diagonal intelligent reflector, characterized in that, include: Obtain the communication management mode of the communication system where the super-diagonal intelligent reflective surface is located; The communication management mode includes either a flexible adjustment mode or a stable convergence mode; Based on the communication management mode being the flexible adjustment mode, the Riemann manifold gradient descent method based on the Lagrange penalty function is used to obtain the integrated waveform control parameters of the syn-sensory system. Based on the stable convergence mode of the communication management mode, the Riemann manifold gradient descent method based on Takagi decomposition is used to obtain the joint waveform control parameters of the syn-inductive integration. The integrated sensing waveform control parameters are transmitted to the impedance matching network of the base station's transmitting radio frequency front end and BD-RIS to control the integrated sensing waveform of the integrated sensing signal and realize the transmission of the integrated sensing signal.

2. The integrated sensing waveform control method based on a super-diagonal intelligent reflector as described in claim 1, characterized in that, The Riemannian manifold gradient descent method based on the Lagrange penalty function is used to obtain the joint waveform control parameters for synesthetic integration, including: The active transmit beam matrix of the base station, the reflection phase shift matrix of BD-RIS, and the power scaling factor are randomly generated in the communication system, and a copy of the reflection phase shift matrix is ​​obtained at the same time. Based on the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the copy of the reflection phase shift matrix, and the power scaling factor, obtain a first joint optimization function with an equality constraint penalty between the reflection phase shift matrix and its copy; Based on the first joint optimization function, the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, a copy of the reflection phase shift matrix, and the first Euclidean gradient corresponding to the power scaling factor are obtained respectively. Based on the first Euclidean gradient, the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the copy of the reflection phase shift matrix, and the first Riemann gradient corresponding to the power scaling factor are obtained respectively. The active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the copy of the reflection phase shift matrix, and the power scaling factor are used as variables. The search is performed along the direction of the first Riemann gradient descent until the first Riemann gradient converges. The updated variables are then projected back onto the manifold using the backtracking operator to obtain the updated active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the copy of the reflection phase shift matrix, and the power scaling factor. If the number of iterations has reached the maximum number of iterations, then obtain the integrated waveform control parameters of the induction system based on the updated active transmit beam matrix and the updated BD-RIS reflection phase shift matrix; otherwise, obtain the residual data based on the updated BD-RIS reflection phase shift matrix and a copy of the updated reflection phase shift matrix, and adaptively adjust the first joint optimization function based on the residual data to obtain the adjusted first joint optimization function. Based on the adjusted first joint optimization function, return to the step of obtaining the first Euclidean gradient and proceed to the next iteration.

3. The integrated sensing waveform control method based on a super-diagonal intelligent reflective surface according to claim 2, characterized in that, Based on the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the copy of the reflection phase shift matrix, and the power scaling factor, the first joint optimization function with the equality constraint penalty between the reflection phase shift matrix and its copy is obtained as follows: ； ； ； ； = ； = ； ； in, Denotes the first joint optimization function. Indicates the weighting coefficient. This represents the radar beam fitting error. This indicates interference error in multi-user communication. Indicates the penalty factor; The reflection phase shift matrix is ​​represented by an off-diagonal matrix, and the response of BD-RIS is characterized by this matrix. as well as ; Let A represent the identity matrix of order A, where A represents the number of elements in the BD-RIS matrix. This represents a copy of the reflection phase shift matrix. The dual variable representing the reflection phase shift matrix, Indicates the sampling point number in the coverage area. Indicates the total number of sampling points. Indicates the directional weighting coefficient. Indicates will Expand and stack them column-wise into a vector. Indicates the active transmit beam matrix, superscript This indicates the conjugate transpose. This represents the radar information matrix characterizing the spatial power distribution. Indicates the power scaling factor. This represents the expected radiated power in the direction of the k-th sampling point. Denotes an L-order identity matrix. Indicates the Kronecker product, superscript The superscript T indicates conjugation, and the superscript T indicates transpose. The guide vector represents a uniform linear array. Represents the natural constant. Represents the imaginary number symbol, This represents the k-th angle. Indicates the number of antennas. This represents the extended equivalent synthesized channel matrix. Indicates will Expand and stack them column-wise into a vector. Represents the desired communication symbol matrix. This represents the equivalent composite channel, where N represents the number of users and L represents the total number of discrete-time snapshots. Represents the field of complex numbers. This refers to the direct link channel from the base station to the user. This represents the reflection link channel between the user and BD-RIS. This indicates the link channel between BD-RIS and the base station.

4. The integrated sensing waveform control method based on a super-diagonal intelligent reflective surface according to claim 3, characterized in that, Based on the first joint optimization function, the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, a copy of the reflection phase shift matrix, and the first Euclidean gradient corresponding to the power scaling factor are obtained, including: ； ； ； ； in, This represents the first Euclidean gradient of the active transmit beam matrix. This represents the first Euclidean gradient of the reflection phase shift matrix of BD-RIS. Let represent the first Euclidean gradient of the copy of the reflection phase shift matrix. This represents the first Euclidean gradient of the power scaling factor. This represents the detection power at the k-th angle. Indicates the discrete-time snapshot number. This represents the total number of discrete-time snapshots. Represents the active transmit beam matrix The column components, The desired communication symbol matrix S represents the first... Column components.

5. The integrated sensing waveform control method based on a super-diagonal intelligent reflector surface according to claim 4, characterized in that, Based on the first Euclidean gradient, the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, a copy of the reflection phase shift matrix, and the first Riemann gradient corresponding to the power scaling factor are obtained, including: ； ； ； = ； in, This represents the first Riemann gradient of the active transmit beam matrix. This represents the first Riemann gradient of the reflection phase shift matrix of BD-RIS. This represents the first Riemann gradient of a copy of the reflection phase shift matrix. Indicates taking the real part, It represents the Hadamardi (or Hadama) stack.

6. The integrated sensing waveform control method based on a super-diagonal intelligent reflective surface according to claim 5, characterized in that, Using the active transmit beam matrix, the BD-RIS reflection phase shift matrix, a copy of the reflection phase shift matrix, and the power scaling factor as variables, a search is performed along the direction of the first Riemann gradient descent until the first Riemann gradient converges. The updated variables are then projected back onto the manifold using a backtracking operator to obtain the updated active transmit beam matrix, the BD-RIS reflection phase shift matrix, a copy of the reflection phase shift matrix, and the power scaling factor, including: The active transmit beam matrix, the reflection phase shift matrix of BD-RIS, a copy of the reflection phase shift matrix, and the power scaling factor are used as variables to search along the direction of the first Riemann gradient descent. The search is repeated until the first Riemann gradient converges. The updated variables are then projected back onto the manifold using the backtracking operator to obtain the updated active transmit beam matrix, the BD-RIS reflection phase shift matrix, a copy of the reflection phase shift matrix, and the power scaling factor: ； ； ； ； in, This represents the active transmit beam matrix after being updated during the h-th search along the direction of the first Riemann gradient descent. This represents the active transmit beam matrix after being updated during the (h-1)th search along the direction of the first Riemann gradient descent. This indicates that the columns are stacked to form a vector. This indicates the step size selected according to the Armijo criterion. Let f(x) represent the reflection phase shift matrix of BD-RIS after the h-th search along the direction of the first Riemann gradient descent. This represents the reflection phase shift matrix of the BD-RIS after being updated during the (h-1)th search along the direction of the first Riemann gradient descent. This represents a copy of the reflection phase shift matrix after the update during the h-th search along the direction of the first Riemann gradient descent. This represents a copy of the reflection phase shift matrix after the update during the (h-1)th search along the direction of the first Riemann gradient descent. This indicates a heavy traction operation, and , This represents the variable matrix in the heavy traction operation. ; To pass through the variable matrix The first singular value matrix obtained by performing singular value decomposition. To pass through the variable matrix The second singular value matrix obtained by performing singular value decomposition. Represents a singular value matrix. This represents the power scaling factor updated during the h-th search along the direction of the first Riemann gradient descent. This represents the updated power scaling factor during the (h-1)th search along the direction of the first Riemann gradient descent.

7. The integrated sensing waveform control method based on a super-diagonal intelligent reflector as described in claim 6, characterized in that, Based on the updated BD-RIS reflection phase shift matrix and a copy of the updated reflection phase shift matrix, residual data is obtained, and the first joint optimization function is adaptively adjusted based on the residual data to obtain the adjusted first joint optimization function, including: Based on the reflection phase shift matrix of the BD-RIS and a copy of the reflection phase shift matrix, the residual data is obtained as follows: ； in, Let represent the reflection phase shift matrix of BD-RIS after the update in the t-th iteration. This represents a copy of the reflection phase shift matrix after the update during the t-th iteration. Denotes the Frobenius norm. This represents the residual data during the t-th iteration. If the residual data is greater than or equal to a preset residual threshold, the penalty factor in the first joint optimization function is updated as follows: ； in, This represents the penalty factor in the t-th iteration. This represents the penalty factor in the (t+1)th iteration, i.e., the penalty factor after the update; This represents the attenuation coefficient between (0,1); If the residual data is less than a preset residual threshold, then the dual variable in the first joint optimization function is updated: ； in, Let represent the dual variable in the t-th iteration. Let represent the dual variable in the (t+1)th iteration, i.e., the dual variable after the update; The first joint optimization function with the updated penalty factor or dual variable is used as the adjusted first joint optimization function.

8. The integrated sensing waveform control method based on a super-diagonal intelligent reflector surface according to claim 5, characterized in that, The joint waveform control parameters for synesthesia integration are obtained using the Riemannian manifold gradient descent method based on Takagi decomposition, including: Randomly generate the active transmit beam matrix of the base station, the reflection phase shift matrix of BD-RIS, and the power scaling factor in the communication system; The reflection phase shift matrix of the BD-RIS is decomposed into a product of unitary matrices according to the Takagi decomposition principle: ;in, This represents the reflection phase shift matrix of BD-RIS. This represents an auxiliary matrix, with the superscript T indicating transpose. Based on the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, and the power scaling factor, the second joint optimization function for fusing the radar beam fitting weighted mean square error and the multi-user communication interference error is obtained as follows: ; ; = ; The second Euclidean gradients corresponding to the active transmit beam matrix, power scaling factor, and auxiliary matrix are obtained according to the second joint optimization function, and the second Riemann gradients corresponding to the active transmit beam matrix, auxiliary matrix, and power scaling factor are obtained according to the second Euclidean gradients. The search is performed along the second Riemann gradient descent direction to obtain the active transmit beam matrix and auxiliary matrix after the search, and the integrated waveform control parameters of the induction system are obtained based on the active transmit beam matrix and auxiliary matrix after the search.

9. The integrated sensing waveform control method based on a super-diagonal intelligent reflector as described in claim 8, characterized in that, Based on the second joint optimization function, the second Euclidean gradients corresponding to the active transmit beam matrix, the reflection phase shift matrix of BD-RIS, the power scaling factor, and the auxiliary matrix are obtained respectively. Based on the second Euclidean gradients, the second Riemann gradients corresponding to the active transmit beam matrix, the auxiliary matrix, and the power scaling factor are obtained respectively, including: Based on the second joint optimization function, the second Euclidean gradients corresponding to the active transmit beam matrix, power scaling factor, and auxiliary matrix are obtained as follows: ； ； ； ； in, This represents the second Euclidean gradient of the active transmission beam matrix. This represents the second Euclidean gradient of the reflection phase shift matrix of BD-RIS. This represents the second Euclidean gradient of the power scaling factor. This represents the second Euclidean gradient of the auxiliary matrix; Based on the second Euclidean gradient, the second Riemann gradients corresponding to the active transmit beam matrix, auxiliary matrix, and power scaling factor are obtained as follows: ； ； = ； in, This represents the second Riemann gradient of the auxiliary matrix. This represents the second Riemann gradient of the active transmission beam matrix. This represents the second Riemann gradient of the power scaling factor. This represents the symmetry operator.

10. The integrated sensing waveform control method based on a super-diagonal intelligent reflector as described in claim 8, characterized in that, A search is performed along the second Riemann gradient descent direction to obtain the searched active transmit beam matrix and auxiliary matrix. Based on the searched active transmit beam matrix and auxiliary matrix, the integrated sensing waveform control parameters are obtained, including: The search proceeds along the descent direction of the second Riemann gradient until the second Riemann gradient converges. The updated variables are then projected back onto the manifold using the backtracking operator, yielding the active transmit beam matrix and auxiliary matrix after the search: ； ； ； in, Indicates the first The active transmission beam matrix during the search process along the second Riemann gradient descent direction, i.e., the active transmission beam matrix after the search. Indicates the current search count. Indicates the first The active transmission beam matrix during the search process along the second Riemann gradient descent direction for the -1st time. Let represent the auxiliary matrix used in the t-th search along the second Riemann gradient descent direction. Indicates the first -1 auxiliary matrix during the search process along the second Riemann gradient descent direction. This represents the extraction of orthogonal matrix factors in the QR decomposition of the matrix. Represents the updated matrix , This represents the orthogonal matrix obtained after performing QR decomposition on V; The optimal active transmission beam matrix is ​​obtained based on the searched active transmission beam matrix. The optimal reflection phase shift matrix is ​​obtained based on the auxiliary matrix after the search. The optimal active transmission beam matrix and the optimal reflection phase shift matrix are used together as the integrated waveform control parameters for the sensing system; wherein, This represents the auxiliary matrix after the search, i.e., the optimal auxiliary matrix.

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