A Phase Shift and Transmit Covariance Matrix Design Method for Stacked BD-RIS-Assisted MIMO Systems

Abstract

The present invention discloses a method for designing phase shift and transmission covariance matrices for a stacked BD-RIS-assisted MIMO system, comprising the following steps: initializing the phase shift matrix and the transmission covariance matrix; optimizing the transmitting-end phase shift matrix using an iterative optimization method; optimizing the receiving-end phase shift matrix using an iterative optimization method based on the optimized transmitting-end phase shift matrix; optimizing the transmission covariance matrix based on the optimized transmitting-end and receiving-end phase shift matrices; and finally repeating the above three steps until the change in system capacity between the two times is less than a threshold, ultimately obtaining the optimal phase shift matrix and transmission covariance matrix. Compared to the stacked RIS structure, the present invention can significantly improve the performance of signal processing in the wave domain by flexibly introducing connections between element-surface components, thereby effectively increasing the overall system capacity.

Description

Technical Field

[0001] The present invention relates to the technical field of wireless communication multi-antenna transmission, and in particular to a method for designing a phase shift and a transmission covariance matrix of a stacked BD-RIS assisted MIMO system. Background Art

[0002] Traditional mobile communication system optimization design primarily focuses on improving system performance at the transmitter and receiver ends. However, the wireless communication environment is limited by factors such as information transmission uncertainty, making it difficult to adjust. This limits the ability to optimize the design of traditional transmit and receive systems, hindering effective system performance improvement. The emergence of RIS technology has opened up new possibilities for optimizing the transmission environment of communication systems. However, the single connection structure of traditional RIS phase elements limits its performance, which in turn affects its beamforming capabilities and flexibility. To overcome these limitations, the research community has begun exploring RIS technologies beyond diagonal matrix, namely BD-RIS. BD-RIS overcomes the limitations of diagonal matrices by introducing mutual coupling between phase elements, allowing the construction of more complex scattering matrices. Consequently, BD-RIS can not only adjust the phase of the incident wave but also distribute and adjust its amplitude across different phase-shifting elements, providing additional performance enhancements. The paper "A low-complexity beamforming design for beyond-diagonal RIS aided multi-user networks" (IEEE Communications Letters, Vol. 28, No. 1, January 2024) derives a closed-form solution for maximizing the receive channel gain when BD-RIS assists multiple single-antenna users. However, when BD-RIS is applied to multi-antenna user scenarios, the design complexity of the phase-shift matrix increases significantly. How to efficiently design the phase-shift matrix to fully utilize the performance advantages of BD-RIS remains a key issue that requires in-depth research and resolution.

[0003] Furthermore, most current RIS research is based on the assumption of a single-layer metasurface, which limits the freedom of beam pattern adjustment and makes it difficult to support more complex signal processing functions. Against this backdrop, researchers have recently proposed the concept of SIM. SIM consists of multiple programmable transmissive metasurfaces, each containing numerous low-cost phase-responsive elements. By appropriately configuring the electromagnetic responses of these elements, SIM can perform more complex signal processing tasks directly in the wave domain. The paper "Stacked intelligent metasurfaces for efficient holographic MIMO communications in 6G" (IEEE Journal on Selected Areas in Communications, Vol. 41, No. 8, August 2023) implements precoding and decoding in the wave domain for MIMO systems, effectively reducing processing latency, hardware complexity, and energy consumption. However, the performance of this approach is limited by the diagonal structure of the RIS, making it difficult to achieve ideal performance, especially when the number of antennas increases. Therefore, if the phase shift matrix and transmit covariance matrix can be jointly designed for stacked BD-RIS-assisted MIMO communication systems, the performance potential of BD-RIS can be fully unleashed, leading to more efficient signal processing in the wave domain and improving overall system performance. Summary of the Invention

[0004] The present invention aims to overcome the shortcomings of the prior art by providing a method for designing phase shift and transmit covariance matrices for a stacked BD-RIS-assisted MIMO system. This method leverages the multi-layer structure and inter-element connectivity of the stacked BD-RIS to flexibly control the phase and amplitude of the incident signal, significantly improving signal processing capabilities in the wave domain. Furthermore, the method optimizes the design of complex super-diagonal phase shift matrices and transmit covariance matrices based on system capacity maximization.

[0005] In order to achieve the above technical objectives, the technical solution adopted by the present invention is:

[0006] The present invention discloses a method for designing phase shift and transmission covariance matrices of a stacked BD-RIS assisted MIMO system, the method comprising the following steps:

[0007] S1, let the sending covariance matrix , is the total transmit power, express The identity matrix, Indicates the number of antennas of the transmitting base station; The phase shift matrix representing the group connection structure is initialized , Indicates the Layer stacked phase shift matrix, express The identity matrix, is the number of reflection units in each BD-RIS layer;

[0008] S2, design the phase shift matrix at the transmitter;

[0009] S3, based on the designed transmitter phase shift matrix, design the receiver phase shift matrix;

[0010] S4, based on the designed transmitter and receiver phase shift matrices, design the transmit covariance matrix;

[0011] S5, repeating steps S2 to S4 until the change in system capacity between the two times is less than a second preset capacity change threshold, thereby obtaining an optimal phase shift matrix and a corresponding user transmit covariance matrix;

[0012] The design process of the phase shift matrix at the transmitting end and the phase shift matrix at the receiving end includes:

[0013] The first The first layer in BD-RIS Group phase shift matrix After decomposition, the corresponding complex unitary matrix is ​​obtained ;based on Projection update of conjugate gradient direction in unitary space , and update Middle Group phase shift matrix ;

[0014] Using the updated Group phase shift matrix and the transmit covariance matrix Update parameters and :

[0015] ;

[0016] ;

[0017] Where, Indicates the stacked BD-RIS matrix at the sending end; Indicates the stacked BD-RIS matrix at the receiving end; is the noise variance of the signal at the receiving end; and Denote the autocorrelation matrix of the base station transmitting and receiving antennas respectively, where express A matrix whose elements come from the set of complex numbers ; Finally, the updated Substitute the next function of system capacity;

[0018] Repeat the above steps until the increment of the next function of this iteration is less than the increment threshold of the next function, and output ;

[0019] according to Different values ​​of , update each set of phase shift matrices of this iteration in turn To design a phase shift matrix ;Will Substitute the large system approximate expression of ergodic capacity to obtain the system capacity;

[0020] Repeat the above steps until the increase in system capacity is less than the first preset capacity change threshold, and output the Phase shift matrix of layer BD-RIS;

[0021] according to For different values ​​of , each phase shift matrix of stacked BD-RIS is designed, where hour, Corresponding to the sending end phase shift matrix; hour, Corresponding to the receiving end phase shift matrix, The number of stacked BD-RIS layers equipped for the transmitting and receiving base stations.

[0022] Step S2 further comprises:

[0023] S21, let the transmitting end group connection structure phase shift matrix , :

[0024] ;

[0025] in, Indicates the number of groups of reflection units on BD-RIS, ; express The identity matrix of represents the matrix conjugate transpose operation, Indicates the matrix transpose operation; use Takagi decomposition to The first layer in the BD-RIS of the sending end Group phase shift matrix Decompose into ,in represents a complex unitary matrix, according to The conjugate gradient direction ,

[0026] ;

[0027] Among them, the intermediate variable matrix ; 、 、 and They are expressed as follows:

[0028] ;

[0029] in Indicates the matrix square root operation; , Represents matrix inversion operation; The BD-RIS phase shift matrix at the transmitter is represented by function, where Indicates the last iteration update Layer optimal phase shift matrix; intermediate variable matrix and Divided into columns Group, and Divided into rows Group; 、 、 and Respectively represent the corresponding Group intermediate variable matrix;

[0030] S22, according to Projection of the conjugate gradient direction in unitary space renew for:

[0031] ;

[0032] in represents the update step size, ;

[0033] renew Middle The group phase shift matrix is:

[0034] ;

[0035] S23, based on 、 And initialized and , update parameters and ; Then substitute the updated parameters into the following formula:

[0036] ;

[0037] in Indicates the system capacity with respect to the sending end The next function of represents taking the real part of a complex scalar, represents finding the trace of the matrix, Indicates finding the Frobeniu norm of the matrix;

[0038] S24, repeat steps S21 to S23 until the , represents the phase shift matrix of the last update, is the incremental threshold of the next function;

[0039] S25, according to Repeat steps S21 to S24 for different values ​​of Design Phase shift matrix of BD-RIS at the layer transmitter ;Will Substitute the large system approximate expression of ergodic capacity to obtain the system capacity:

[0040] ;

[0041] in Indicates finding the determinant of a matrix;

[0042] S26, repeat steps S21 to S25, iteratively update Until the increase in system capacity is less than the first preset capacity change threshold , output the best ;

[0043] S27, according to For different values ​​of , steps S21 to S26 are repeated to design each phase shift matrix of the stacked BD-RIS at the transmitting end.

[0044] Step S3 further comprises:

[0045] S31, let the receiving end group connection structure phase shift matrix , :

[0046] ;

[0047] in, Indicates the number of groups of reflection units on BD-RIS, ; Use Takagi decomposition to The first layer in the BD-RIS of the sending end Group phase shift matrix Decompose into ,in represents a complex unitary matrix; obtain The conjugate gradient direction is :

[0048] ;

[0049] Among them, the intermediate variable matrix ; 、 、 and They are expressed as follows:

[0050] ;

[0051] in ; The stacked BD-RIS phase shift matrix at the receiving end is represented by function, where Indicates the last iteration update Layer optimal phase shift matrix; intermediate variable matrix and Divided into columns Group, and Divided into rows Group; 、 、 and Indicates the first Group intermediate variable matrix;

[0052] S32, according to Projection of the conjugate gradient direction in unitary space renew for:

[0053] ;

[0054] in represents the update step size, ;

[0055] renew Middle The group phase shift matrix is:

[0056] ;

[0057] S33, based on 、 And initialized and Update parameters and , and then substitute the updated parameters into the following formula:

[0058] ;

[0059] in Indicates the system capacity with respect to the receiving end The next function of

[0060] S34, repeat step S33 to step S33 until the , represents the phase shift matrix updated in the last iteration, is the incremental threshold of the next function;

[0061] S35, according to Repeat steps S31 to S34 for different values ​​of To design a phase shift matrix , based on the design Obtain system capacity;

[0062] S36, repeat steps S31 to S35, iteratively update Until the increase in system capacity is less than the first preset capacity change threshold , output the best ;

[0063] S37, according to Repeat steps S31 to S36 for different values ​​of to design each phase shift matrix of the stacked BD-RIS at the receiving end.

[0064] Furthermore, the parameter and It is obtained through iterative calculation, and the calculation process includes:

[0065] initialization and ,Will Substitution beg , and then Backward Substitution beg ; Repeat until and Converges to the preset threshold.

[0066] Step S4 further comprises:

[0067] S41, yes Do eigenvalue decomposition:

[0068] ;

[0069] yes The eigenvector matrix of It is composed of eigenvalues The diagonal matrix of ;

[0070] S42, given the optimal power allocation matrix Before that, you need to calculate the parameters , this parameter can be obtained by solving the following The one-variable equation is obtained:

[0071] ;

[0072] in It takes a positive sign, which means that all negative elements in the sign are replaced by 0. express The element in row i and column i;

[0073] S43, design and obtain the optimal transmission covariance matrix :

[0074] ;

[0075] S44, based on and , update parameters and .

[0076] Step S5 further comprises:

[0077] S51, based on the updated step S4 、 、 and Calculate the system capacity;

[0078] S52, repeat steps S2 to S4 until the overall gain of the system capacity is less than the second preset capacity change threshold , output the optimal phase shift matrix and the optimal transmit covariance matrix ;

[0079] S53, recalculate parameters based on the optimal phase shift matrix and the optimal transmit covariance matrix and , calculate the final system capacity.

[0080] Compared with the prior art, the present invention has the following beneficial effects:

[0081] First, the phase shift and transmit covariance matrix design method of the stacked BD-RIS assisted MIMO system of the present invention fully utilizes statistical channel state information during the stacked BD-RIS assisted MIMO communication process and jointly designs the transmit covariance matrix and phase shift matrix. Compared with the design method based on instantaneous channel state information, the method proposed by the present invention has lower requirements for prior channel knowledge and does not require real-time signal processing. While further improving system performance, it also reduces hardware requirements and cost overhead.

[0082] Second, the present invention's phase shift and transmit covariance matrix design method for a stacked BD-RIS-assisted MIMO system overcomes the limitations of conventional stacked RIS diagonal structures by introducing stacked BD-RIS structures at both the transmitter and receiver for wave-domain signal processing. This design not only flexibly controls the phase of the incident signal but also precisely adjusts the distribution of the signal amplitude across different phase-shifting elements, significantly improving signal processing capabilities in the wave domain and providing strong support for achieving more efficient communication systems.

[0083] Third, the present invention addresses the phase shift and transmit covariance matrix design methods for a stacked BD-RIS-assisted MIMO system. Since the optimization methods for the phase shift matrix in a stacked RIS are no longer applicable, the present invention addresses the interconnection characteristics of the stacked BD-RIS by designing an iterative optimization method that accommodates different connection structures. Specifically, the present invention constructs lower-bound functions for different connection structures and progressively optimizes these lower-bound functions through an iterative process. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] Figure 1 Flowchart of the phase shift and transmit covariance matrix design method for the stacked BD-RIS assisted MIMO system of the present invention;

[0085] Figure 2 It is a convergence result diagram of the iterative algorithm of the present invention;

[0086] Figure 3 Schematic diagram of simulation results of an embodiment of the present invention. DETAILED DESCRIPTION

[0087] The embodiments of the present invention are described in further detail below with reference to the accompanying drawings.

[0088] The present invention proposes a method for designing the phase shift and transmit covariance matrices for a stacked BD-RIS-assisted MIMO system. First, the transmit covariance matrix and phase shift matrix are initialized. Next, the transmit phase shift matrix is ​​designed. Next, based on the designed transmit phase shift matrix, the receive phase shift matrix is ​​designed. Then, based on the designed transmit and receive phase shift matrices, the transmit covariance matrix is ​​designed. Finally, the previous three steps are repeated until the system capacity converges to a threshold. The present invention effectively utilizes statistical channel state information, avoiding the real-time estimation of channel state information. Based on the inter-component connection characteristics of BD-RIS, the phase shift matrix and transmit covariance matrix are jointly optimized, effectively improving system performance.

[0089] The present invention will be further described in detail below with reference to the accompanying drawings:

[0090] like Figure 1 As shown, the present invention proposes a method for designing phase shift and transmission covariance matrix of a stacked BD-RIS assisted MIMO system. The stacked BD-RIS assisted MIMO communication system includes two and The transmitting and receiving base stations of the root antenna are equipped with and Layer stacking BD-RIS, without loss of generality, in the subsequent steps let , each layer of BD-RIS has The base station has known statistical channel state information, that is, the base station transmitting end antenna autocorrelation matrix and the autocorrelation matrix of the user receiving antenna ,in express A matrix whose elements come from the set of complex numbers ,In addition, the phase shift matrix of BD-RIS is a group connection structure that can be flexibly adjusted;

[0091] Step 1. Let the send covariance matrix , is the total transmit power, express The identity matrix of The phase shift matrix representing the group connection structure is initialized .in Indicates the Layer stacked phase shift matrix, When , it represents the phase shift matrix at the transmitting end; When , it represents the phase shift matrix at the receiving end; express The identity matrix of

[0092] Step 2. Design the transmitter phase shift matrix;

[0093] Step 2.1. Transmitter group connection structure phase shift matrix , :

[0094] (1);

[0095] in , represents the number of groups of reflection units on BD-RIS; express The identity matrix of represents the matrix conjugate transpose operation, Indicates the matrix transpose operation. Use Takagi decomposition to The first layer in the BD-RIS of the sending end Group phase shift matrix Decompose into ,in represents a complex unitary matrix. Get The conjugate gradient direction is :

[0096] (2);

[0097] Among them, the intermediate variable matrix ; , , and They are expressed as follows:

[0098] (3);

[0099] in Indicates the matrix square root operation; Indicates the stacked BD-RIS matrix at the sending end; Indicates the Layer to The inter-layer transfer matrix of layer BD-RIS, ; represents the inter-layer transmission matrix from the first layer BD-RIS to the transmitting antenna; , Represents matrix inversion operation; The BD-RIS phase shift matrix at the transmitter is represented by function, where Indicates the last iteration update Layer optimal phase shift matrix. Intermediate variable matrix and Divided into columns Group, and Divided into rows Group; , , and Respectively represent the corresponding Group intermediate variable matrix; and As a set of parameters that appear simultaneously, they are defined as:

[0100] (4);

[0101] (5);

[0102] in Indicates the stacked BD-RIS matrix at the receiving end; Indicates the Layer to The inter-layer transfer matrix of layer BD-RIS, ; Indicates the Inter-layer transmission matrix from layer BD-RIS to receiving antennas. is the noise variance of the signal at the receiving end; and It needs to be obtained through iterative calculation, and the calculation method is:

[0103] initialization , ,Will Substitution beg , then Backward Substitution beg ; Repeat this process until and Converges to a threshold, where the threshold is a very small positive number;

[0104] Step 2.2. According to Projection of the conjugate gradient direction in unitary space renew for:

[0105] (6);

[0106] in represents the update step size, ;

[0107] renew Middle The group phase shift matrix is:

[0108] (7);

[0109] Step 2.3. , , initialized and Substitute into equations (4) and (5) to update the parameters and . Then substitute these parameters into the following formula:

[0110] (8);

[0111] in Indicates the system capacity with respect to the sending end The next function of represents taking the real part of a complex scalar, represents finding the trace of the matrix, Indicates finding the Frobeniu norm of the matrix;

[0112] Step 2.4. Repeat steps 2.1-2.3 until the , represents the phase shift matrix updated in the last iteration, is the increment threshold of the next function, which can be set to a very small positive number;

[0113] Step 2.5. According to Repeat step 2.4 for different values ​​of and update each set of phase shift matrices in turn To design a phase shift matrix .Will Substitute the large system approximate expression of ergodic capacity to obtain the system capacity, as shown below:

[0114] (9);

[0115] in Indicates finding the determinant of a matrix;

[0116] Step 2.6. Repeat step 2.5 until the increase in system capacity is less than the pre-set threshold. , is a very small positive number. At this time, the optimal ;

[0117] Step 2.7. According to Repeat step 2.6 for different values ​​of to design each phase shift matrix of the transmitting-end stacked BD-RIS.

[0118] Step 3. Based on the designed transmitter phase shift matrix, design the receiver phase shift matrix.

[0119] Step 3.1. Receiver group connection structure phase shift matrix , :

[0120] (10);

[0121] in , represents the number of groups of reflection units on BD-RIS. The first layer in the BD-RIS of the sending end Group phase shift matrix Decompose into ,in represents a complex unitary matrix. Get The conjugate gradient direction is :

[0122] (11);

[0123] Among them, the intermediate variable matrix ; , , and They are expressed as follows:

[0124] (12);

[0125] in ; The stacked BD-RIS phase shift matrix at the receiving end is represented by function, where Indicates the last iteration update Layer optimal phase shift matrix; intermediate variable matrix and Divided into columns Group, and Divided into rows Group; , , and Indicates the first Group intermediate variable matrix;

[0126] Step 3.2. According to Projection of the conjugate gradient direction in unitary space renew for:

[0127] (13);

[0128] in represents the update step size, ;

[0129] renew Middle The group phase shift matrix is:

[0130] (14);

[0131] Step 3.3. , , initialized and Substitute into equations (4) and (5) to update the parameters and . Then substitute these parameters into the following formula:

[0132] (15);

[0133] in Indicates the system capacity with respect to the sending end The next function of

[0134] Step 3.4. Repeat steps 3.1 to 3.3 until the , represents the phase shift matrix updated in the last iteration, is the threshold, which can be set to a very small positive number;

[0135] Step 3.5. According to Repeat step 3.4 for different values ​​of , and update each set of phase shift matrices in turn To design a phase shift matrix .Will Substitute into (10) to obtain the system capacity;

[0136] Step 3.6. Repeat step 3.5 and iterate Until the increase in system capacity is less than the pre-set threshold , is a very small positive number. At this time, the optimal ;

[0137] Step 3.7. According to Repeat step 3.6 for different values ​​of to design each phase shift matrix of the receiving-end stacked BD-RIS.

[0138] Step 4. Based on the designed transmitter and receiver phase shift matrices, design the transmit covariance matrix.

[0139] Step 4.1. Do eigenvalue decomposition:

[0140] (16);

[0141] yes The eigenvector matrix of It is composed of eigenvalues The diagonal matrix of ;

[0142] Step 4.2. Given the optimal transmit covariance matrix Before that, you need to calculate the parameters , this parameter can be obtained by solving the following The one-variable equation is obtained:

[0143] (17);

[0144] in It takes a positive sign, which means that all negative elements in the sign are replaced by 0. express The element in row i and column i;

[0145] Step 4.3. Design the optimal transmit covariance matrix :

[0146] (18);

[0147] Step 4.4. Based on and ,in . Substitute into equations (4) and (5) again to update the parameters and .

[0148] Step 5. , , and Substitute back into the system capacity expression in formula (10), and repeat steps 2, 3, and 4 until the overall gain in system capacity is less than the set threshold. The optimization is considered to be completed. At this time, the optimal phase shift matrix is ​​output ,in ;Optimal transmit covariance matrix ; and bring these parameters into equations (4) and (5) to recalculate the two parameters and ; Then substitute it back into formula (10) to calculate the final system capacity.

[0149] Figure 2The figure shows the convergence result of the iterative algorithm of the present invention, where the parameters are set as follows: the transmitting and receiving base stations each have 6 antennas, there are 100 passive components on each layer of stacked BD-RIS, and the number of stacked BD-RIS layers at the transmitting and receiving ends is 3. Figure 2 The horizontal axis is the number of iterations, and the vertical axis is the system capacity, which is expressed in bits per second per Hertz. This means the number of bits transmitted per second by the base station in a bandwidth of 1 Hertz. The four curves in the figure are: represents a fully connected BD-RIS structure, where all elements on each stacking element surface are connected to each other; Represents a group-connected BD-RIS structure, where the elements on each stacking element surface are divided into two groups, with 50 elements in each group connected to each other; It means that the elements on the surface of each stacked element are divided into 4 groups, and each group of 25 elements are connected to each other; The elements on each stacked element surface are divided into 10 groups, each consisting of 10 interconnected elements. As can be seen from the figure, the proposed algorithm converges within 20 iterations, demonstrating excellent convergence performance. Furthermore, as the number of inter-element connections increases, the system's achievable capacity steadily increases. This phenomenon clearly demonstrates that the introduction of inter-element connections can significantly enhance the signal processing capabilities of the stacked BD-RIS in the wave domain, providing strong support for optimizing system performance.

[0150] Figure 3 The figure shows the simulation results of the present invention, wherein the parameters are set as follows: the transmitting and receiving base stations each have 6 antennas, each layer of stacked BD-RIS has 100 passive components, and the number of stacked BD-RIS layers at the transmitting end is fixed to 3. Figure 3 The horizontal axis is the number of BD-RIS layers stacked at the receiving end, and the vertical axis is the system capacity, measured in bits per second per hertz. represents a fully connected BD-RIS structure, where all elements on each stacking element surface are connected to each other; Represents a group-connected BD-RIS structure, where the elements on each stacking element surface are divided into 10 groups, and each group of 10 elements is interconnected; represents a single-connection RIS structure, where no connections exist between the elements on each stacked layer. As can be seen from the figure, as the number of stacked layers increases, stacked BD-RIS demonstrates significant superiority in wave-domain signal processing capabilities compared to stacked RIS. This advantage directly translates into a significant increase in system capacity, demonstrating that BD-RIS can more efficiently utilize wave-domain resources in a multi-layer stacked structure, resulting in significant gains in system performance.

[0151] Although the preferred embodiments of the present application have been described, those skilled in the art may make additional changes and modifications to these embodiments once they have learned the basic creative concept. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments and all changes and modifications that fall within the scope of the present application.

[0152] Obviously, those skilled in the art may make various changes and modifications to this application without departing from the spirit and scope of this application. Thus, if these modifications and variations of this application fall within the scope of the claims of this application and their equivalents, this application is intended to include these modifications and variations.

Claims

1. A method for designing phase shift and transmit covariance matrices for a stacked BD-RIS assisted MIMO system, characterized in that: The method comprises the following steps: S1, let the sending covariance matrix , is the total transmit power, express The identity matrix, Indicates the number of antennas of the transmitting base station; The phase shift matrix representing the group connection structure is initialized , Indicates the Layer stacked phase shift matrix, express The identity matrix, is the number of reflection units in each BD-RIS layer; S2, design the phase shift matrix at the transmitter; S3, based on the designed transmitter phase shift matrix, design the receiver phase shift matrix; S4, based on the designed transmitter and receiver phase shift matrices, design the transmit covariance matrix; S5, repeating steps S2 to S4 until the change in system capacity between the two times is less than a second preset capacity change threshold, thereby obtaining an optimal phase shift matrix and a corresponding user transmit covariance matrix; The design process of the phase shift matrix at the transmitting end and the phase shift matrix at the receiving end includes: The first The first layer in BD-RIS Group phase shift matrix After decomposition, the corresponding complex unitary matrix is ​​obtained ;based on Projection update of conjugate gradient direction in unitary space , and update Middle Group phase shift matrix ; Using the updated Group phase shift matrix and the transmit covariance matrix Update parameters and : ； ； Where, Indicates the stacked BD-RIS matrix at the sending end; Indicates the stacked BD-RIS matrix at the receiving end; is the noise variance of the signal at the receiving end; and Denote the autocorrelation matrix of the base station transmitting and receiving antennas respectively, where express A matrix whose elements come from the set of complex numbers ; Finally, the updated Substitute the next function of system capacity; Repeat the above steps until the increment of the next function of this iteration is less than the increment threshold of the next function, and output ; according to Different values ​​of , update each set of phase shift matrices of this iteration in turn To design a phase shift matrix ;Will Substitute the large system approximate expression of ergodic capacity to obtain the system capacity; Repeat the above steps until the increase in system capacity is less than the first preset capacity change threshold, and output the Phase shift matrix of layer BD-RIS; according to For different values ​​of , each phase shift matrix of stacked BD-RIS is designed, where hour, Corresponding to the sending end phase shift matrix; hour, Corresponding to the receiving end phase shift matrix, The number of stacked BD-RIS layers configured for the transmitting and receiving base stations.

2. The method for designing phase shift and transmission covariance matrix of a stacked BD-RIS assisted MIMO system according to claim 1, wherein: Step S2 further comprises: S21, let the transmitting end group connection structure phase shift matrix , : ； in, Indicates the number of groups of reflection units on BD-RIS, ; express The identity matrix of represents the matrix conjugate transpose operation, Indicates the matrix transpose operation; use Takagi decomposition to The first layer in the BD-RIS of the sending end Group phase shift matrix Decompose into ,in represents a complex unitary matrix, according to The conjugate gradient direction , ； Among them, the intermediate variable matrix ; 、 、 and They are expressed as follows: ； in Indicates matrix square root operation; , Represents matrix inversion operation; The BD-RIS phase shift matrix at the transmitter is represented by function, where Indicates the last iteration update Layer optimal phase shift matrix; intermediate variable matrix and Divided into columns Group, and Divided into rows Group; 、 、 and Respectively represent the corresponding Group intermediate variable matrix; S22, according to Projection of the conjugate gradient direction in unitary space renew for: ； in represents the update step size, ; renew Middle The group phase shift matrix is: ； S23, based on 、 And initialized and , update parameters and ; Then substitute the updated parameters into the following formula: ； in Indicates the system capacity with respect to the sending end The next function of represents taking the real part of a complex scalar, represents finding the trace of the matrix, Indicates finding the Frobeniu norm of the matrix; S24, repeat steps S21 to S23 until the , represents the phase shift matrix of the last update, is the incremental threshold of the next function; S25, according to Repeat steps S21 to S24 for different values ​​of Design Phase shift matrix of BD-RIS at the layer transmitter ;Will Substitute the large system approximate expression of ergodic capacity to obtain the system capacity: ； in Indicates finding the determinant of a matrix; S26, repeat steps S21 to S25, iteratively update Until the increase in system capacity is less than the first preset capacity change threshold , output the best ; S27, according to For different values ​​of , steps S21 to S26 are repeated to design each phase shift matrix of the stacked BD-RIS at the transmitting end.

3. The method for designing phase shift and transmission covariance matrix of a stacked BD-RIS assisted MIMO system according to claim 1, wherein: Step S3 further comprises: S31, let the receiving end group connection structure phase shift matrix , : ； in, Indicates the number of groups of reflection units on BD-RIS, ; Use Takagi decomposition to The first layer in the BD-RIS of the sending end Group phase shift matrix Decompose into ,in represents a complex unitary matrix; obtain The conjugate gradient direction is : ； Among them, the intermediate variable matrix ; 、 、 and They are expressed as follows: ； in ; The stacked BD-RIS phase shift matrix at the receiving end is represented by function, where Indicates the last iteration update Layer optimal phase shift matrix; intermediate variable matrix and Divided into columns Group, and Divided into rows Group; 、 、 and Indicates the first Group intermediate variable matrix; S32, according to Projection of the conjugate gradient direction in unitary space renew for: ； in represents the update step size, ; renew Middle The group phase shift matrix is: ； S33, based on 、 And initialized and Update parameters and , and then substitute the updated parameters into the following formula: ； in Indicates the system capacity with respect to the receiving end The next function of S34, repeat step S33 to step S33 until the , represents the phase shift matrix updated in the last iteration, is the incremental threshold of the next function; S35, according to Repeat steps S31 to S34 for different values ​​of To design a phase shift matrix , based on the design Obtain system capacity; S36, repeat steps S31 to S35, iteratively update Until the increase in system capacity is less than the first preset capacity change threshold , output the best ; S37, according to Repeat steps S31 to S36 for different values ​​of to design each phase shift matrix of the stacked BD-RIS at the receiving end.

4. The method for designing phase shift and transmission covariance matrix of a stacked BD-RIS assisted MIMO system according to claim 1, wherein: Parameters and It is obtained through iterative calculation, and the calculation process includes: initialization and ,Will Substitution beg , and then Backward Substitution beg ; Repeat until and Converges to the preset threshold.

5. The method for designing phase shift and transmission covariance matrix of a stacked BD-RIS assisted MIMO system according to claim 1, wherein: Step S4 further comprises: S41, yes Do eigenvalue decomposition: ； yes The eigenvector matrix of It is composed of eigenvalues The diagonal matrix of ; S42, given the optimal power allocation matrix Before that, you need to calculate the parameters , this parameter can be obtained by solving the following The one-variable equation is obtained: ； in It takes a positive sign, which means that all negative elements in the sign are replaced by 0. express The element in row i and column i; S43, design and obtain the optimal transmission covariance matrix : ； S44, based on and , update parameters and .

6. The method for designing phase shift and transmission covariance matrix of a stacked BD-RIS assisted MIMO system according to claim 5, wherein: Step S5 further comprises: S51, based on the updated step S4 、 、 and Calculate the system capacity; S52, repeat steps S2 to S4 until the overall gain of the system capacity is less than the second preset capacity change threshold , output the optimal phase shift matrix and the optimal transmit covariance matrix ; S53, recalculate parameters based on the optimal phase shift matrix and the optimal transmit covariance matrix and , calculate the final system capacity.

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