Scattering matrix optimization method for maximizing MIMO capacity with super-diagonal reconfigurable intelligent surface assistance
By employing a two-layer iterative optimization algorithm combining augmented Lagrange and Riemann conjugate gradients, the constraint problem of super-diagonal reconfigurable smart surfaces during the optimization process was solved, resulting in a significant improvement in the capacity of MIMO systems.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- ANHUI NORMAL UNIV
- Filing Date
- 2026-04-29
- Publication Date
- 2026-07-14
AI Technical Summary
Superdiagonal reconfigurable smart surfaces face more constraints during the optimization process, which increases the difficulty of optimization and makes it difficult for existing technologies to effectively improve capacity.
A two-level iterative optimization algorithm combining augmented Lagrangian and Riemann conjugate gradients is adopted. The constraints are simplified by constructing an augmented Lagrangian function, and the Riemann conjugate gradient method is used to optimize on the unitary matrix manifold, ensuring that the unitary constraints are satisfied during the iteration process.
It significantly improves the capacity of the super-diagonal reconfigurable smart surface-assisted MIMO system, especially under high transmit power conditions, and outperforms traditional methods.
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Abstract
Description
Technical Field
[0001] This invention relates to the field of wireless communication technology, and mainly to a scattering matrix optimization method for maximizing the capacity of superdiagonal reconfigurable smart surface-assisted MIMO. Background Technology
[0002] With the rapid development of 6G wireless communication technology, reconfigurable smart surfaces have become an emerging paradigm. Superdiagonal reconfigurable smart surfaces, as a novel type of reconfigurable smart surface, have been proposed, offering higher degrees of freedom, capacity, and beamforming gain. However, superdiagonal reconfigurable smart surfaces face more constraints, making their optimization more challenging. Therefore, this invention proposes a two-layer iterative optimization algorithm combining augmented Lagrangian and Riemann conjugate gradients for optimizing the scattering matrix of superdiagonal reconfigurable smart surfaces. Summary of the Invention
[0003] The purpose of this invention is to address the shortcomings of existing technologies by proposing a scattering matrix optimization method for maximizing the capacity of superdiagonally reconfigurable smart surfaces-assisted MIMO. This method includes the following steps:
[0004] Step 1: Definition and The new equivalent channel can be expressed as:
[0005]
[0006] Step 2: The capacity maximization problem for only the superdiagonally reconfigurable smart surface (BD-RIS) can be expressed as:
[0007]
[0008] Step 3: Construct the augmented Lagrangian function as follows:
[0009]
[0010] in To handle symmetric-constrained Lagrange multipliers, Y is the Lagrange multiplier matrix. It is a residual matrix with symmetric constraints; The penalty term for the augmented Lagrange function is ρ, which is the penalty factor.
[0011] However, optimization of the constructed augmented Lagrangian function is constrained by fixing Y and ρ, and minimizing... Obtain the optimal symmetric matrix Θ, and then update the Lagrange multipliers:
[0012]
[0013] Increase the penalty factor:
[0014]
[0015] In the formula, η is the penalty factor amplification coefficient. Finally, convergence is judged if the residuals satisfy:
[0016]
[0017] Then stop updating; at this point, Θ is the optimal solution.
[0018] Step 4: After constructing the augmented Lagrangian function, only one unitary constraint remains, and the set of unitary matrices can themselves form a smooth manifold. The tangent space of the manifold at point Q is:
[0019]
[0020] Therefore, the Riemann conjugate gradient method can be used to optimize the manifold and find the optimal Θ for the constructed augmented Lagrangian function.
[0021] For augmented Lagrangian functions Taking the partial derivative with respect to Θ, we obtain the Euclidean gradient. Projecting the Euclidean gradient onto the tangent space of the unitary manifold at point Q yields the Riemannian gradient. ):
[0022]
[0023] The projection operation ensures that the gradient direction lies within the tangent space, which conforms to the geometric constraints of manifold optimization.
[0024] The first search direction of the Riemann conjugate gradient method is the negative of the Riemann gradient:
[0025]
[0026] In subsequent iterations, the search direction of the Riemann conjugate gradient method is formed by the conjugate combination of the current Riemann gradient and the previous search direction. The search direction in the k-th iteration is... for:
[0027]
[0028] Wherein, conjugate coefficient and search direction Both require transporting the old vector to the new tangent space to complete the computation:
[0029]
[0030]
[0031]
[0032]
[0033] In search direction The optimal step size is found using Armijo line search. ,satisfy:
[0034]
[0035] in, It is a slanted Hermitian matrix. For matrix exponents, For Armijo line search coefficients.
[0036] The update of the unitary matrix uses an exponential mapping to ensure that the updated matrix still belongs to the unitary matrix manifold:
[0037]
[0038] Exponential mapping is a core operation on Riemannian manifolds, realizing a structure-preserving mapping from the tangent space to the manifold and ensuring that the unitary constraint is always satisfied during the iteration process.
[0039] Repeat the above steps until the norm of the Riemann gradient satisfies ( (As the gradient convergence threshold), the optimal unitary matrix under the current augmented Lagrange method is obtained. .
[0040] This invention innovatively proposes a two-layer iterative optimization method for the dual-constraint optimization problem of superdiagonal reconfigurable smart surfaces, which optimizes the dual constraints separately and improves the capacity compared with existing technologies. Attached Figure Description
[0041] Figure 1 Diagram of a super-diagonal reconfigurable smart surface system
[0042] Figure 2 These are curves showing the achievable rate versus transmit power under superdiagonal RIS and RIS architectures.
[0043] Figure 3 The rate of increase of superdiagonal RIS with increasing number of cells M
[0044] Figure 4 It is the achievable speed of a 100-unit superdiagonal RIS-assisted 4×4 MIMO link. Detailed Implementation
[0045] The following is a detailed description of the present invention:
[0046] This embodiment considers a BD-RIS-assisted MIMO communication link, and N is configured at the BS transmitter in Figure 1. T One antenna, UE receiver configured with N R The system has one antenna, a superdiagonal RIS containing M fully connected reflector elements, and both the transmitter and receiver acquire perfect channel state information (CSI). The system's equivalent N... R ×N T The MIMO channel is:
[0047]
[0048] in, The channel matrix from the transmitter to BD-RIS. This is the channel matrix from BD-RIS to the receiver. Direct link channel matrix between Let be the scattering matrix of the M×M superdiagonal RIS.
[0049] For a fully connected reflective superdiagonal RIS, its scattering matrix must satisfy unitary and symmetry constraints, and the feasible region is:
[0050]
[0051] in, The unitary constraint ensures that BD-RIS is passive and lossless; As a symmetry constraint, it is determined by the propagation characteristics of reciprocal passive networks, that is, the power loss between any two ports is independent of the propagation direction.
[0052] The transmitter sends a complex Gaussian signal. Among them, satisfying and Based on the above assumptions, the problem of maximizing the capacity of a MIMO link assisted by a superdiagonally reconfigurable smart surface (BD-RIS) can be constructed as follows:
[0053]
[0054]
[0055]
[0056] In the formula Let be the noise covariance matrix of the additive white Gaussian noise at the receiving end. Let be the equivalent channel matrix in the formula. In this embodiment, log represents the natural logarithm, with units of nats.
[0057] The transmitting end is equipped with NT antennas, and the receiving end is equipped with NR antennas, with spatial coordinates of (0,0,1.5) and (50,0,1.5) respectively (all coordinate units are meters). The reconfigurable smart surface (RIS) has coordinates of (d,5,5), where parameter d varies along the x-axis, ranging from 10 meters to 100 meters. The system operates with a bandwidth of 20 MHz and a frequency of 2.4 GHz.
[0058] The path loss model is PL = PL0 − α ⋅ 10log 10 d, where PL0 = −28 dB is the path loss at a reference distance d0 = 1 meter, and α is the path loss exponent. The power spectral density of additive noise is
[0059]
[0060] The unit is dBm; the transmit power P = 100mW.
[0061] For small-scale fading in MIMO direct links, Rayleigh channel modeling is used, while for large-scale fading, the path loss exponent is taken as α. d =3.75; For G and F channels relayed via RIS, the path loss exponent α=2, and the small-scale fading is assumed to be Ricean fading with a Ricean factor γ=3. H, using Rayleigh fading, can be expressed as:
[0062]
[0063] Since F and G are Ricean fading, the channel is divided into line-of-sight (LOS) and non-line-of-sight (NLOS) components. First, the array response vector is defined as:
[0064]
[0065]
[0066] In the formula Given random angles of arrival / departure, channels F and G can be obtained:
[0067]
[0068]
[0069] For solving this type of non-convex problem, this embodiment employs a commonly used alternating optimization method: first, fix Θ, and then optimize the covariance matrix R... xx Optimize; then fix R xx We optimize Θ and iterate repeatedly until convergence.
[0070] Step 1: With Θ fixed, solve for the optimal emission covariance matrix R that maximizes capacity. xxThis is a convex optimization problem, and its solution is:
[0071]
[0072] Where V is the equivalent MIMO channel matrix The left eigenspace matrix is given by P = diag(P1,…,Pd), which is a diagonal matrix. Pi represents the optimal power allocated to the i-th data stream using the water injection power allocation strategy, which satisfies the power constraint:
[0073]
[0074] Step 2: Obtain the emission covariance matrix R xx After that, it can be defined and R xx Influence Integration and Then, the new equivalent channel can be expressed as:
[0075]
[0076] The channel's dependence on the scattering matrix is specifically emphasized here. The capacity maximization problem for the superdiagonal reconfigurable smart surface (BD-RIS) can be expressed as:
[0077]
[0078] Step 3: In optimizing the capacity maximization problem Θ, Θ has two complex constraints: unitary and symmetric constraints, making optimization difficult. Therefore, an augmented Lagrangian function can be constructed to integrate one of these constraints into the optimization objective, retaining only one constraint to simplify the optimization. In this embodiment, the symmetric constraint is chosen to be integrated into the optimization objective function because its impact is smaller than that of the unitary constraint. Therefore, the augmented Lagrangian function can be constructed as follows:
[0079]
[0080] in To handle symmetric-constrained Lagrange multipliers, Y is the Lagrange multiplier matrix. It is a residual matrix with symmetric constraints; The penalty term for the augmented Lagrange function is ρ, which is the penalty factor.
[0081] However, optimization of the constructed augmented Lagrangian function is constrained by fixing Y and ρ, and minimizing... Obtain the optimal symmetric matrix Θ, and then update the Lagrange multipliers:
[0082]
[0083] Increase the penalty factor:
[0084]
[0085] In the formula, η is the penalty factor amplification coefficient. Finally, convergence is judged if the residuals satisfy:
[0086]
[0087] Then stop updating; at this point, Θ is the optimal solution.
[0088] Step 4: After constructing the augmented Lagrangian function, only one unitary constraint remains, and the set of unitary matrices can themselves form a smooth manifold. The tangent space of the manifold at point Q is:
[0089]
[0090] The tangent space consists of matrices satisfying the skew Hermitian condition, with a real dimension of M². The core of the Riemann conjugate gradient method is to construct conjugate search directions within the tangent space to achieve efficient gradient descent on the manifold.
[0091] Therefore, the Riemann conjugate gradient method can be used to optimize the manifold and find the optimal Θ for the constructed augmented Lagrangian function. The Riemann conjugate gradient method consists of four steps:
[0092] Step 5; For the augmented Lagrangian function Taking the partial derivative with respect to Θ, we obtain the Euclidean gradient. Projecting the Euclidean gradient onto the tangent space of the unitary manifold at point Q yields the Riemannian gradient. :
[0093]
[0094] The projection operation ensures that the gradient direction lies within the tangent space, which conforms to the geometric constraints of manifold optimization.
[0095] Step Six: The first search direction of the Riemann conjugate gradient method is the negative of the Riemann gradient:
[0096]
[0097] In subsequent iterations, the search direction of the Riemann conjugate gradient method is formed by the conjugate combination of the current Riemann gradient and the previous search direction. The search direction in the k-th iteration is... for:
[0098]
[0099] Wherein, conjugate coefficient and search direction Both require transporting the old vector to the new tangent space to complete the computation:
[0100]
[0101]
[0102]
[0103]
[0104] Step 7: In the search direction The optimal step size is found using Armijo line search. ,satisfy:
[0105]
[0106] in, It is a slanted Hermitian matrix. For matrix exponents, For Armijo line search coefficients.
[0107] The update of the unitary matrix uses an exponential mapping to ensure that the updated matrix still belongs to the unitary matrix manifold:
[0108]
[0109] Exponential mapping is a core operation on Riemannian manifolds, realizing a structure-preserving mapping from the tangent space to the manifold and ensuring that the unitary constraint is always satisfied during the iteration process.
[0110] Step 8: Repeat the above steps until the norm of the Riemann gradient satisfies ( (As the gradient convergence threshold), the optimal unitary matrix under the current augmented Lagrange method is obtained. .
[0111] As shown in Figure 2, the achievable data rates of the system are as follows: in a 100-element BD-RIS-assisted 4×4 and 8×8 MIMO link, the input power varies between 4 and 30 dBm (corresponding to a receiver signal-to-noise ratio of approximately 15 to 35 dB). The performance improvement of the algorithm proposed in this embodiment is more significant at higher transmit powers, and the improvement is even more pronounced in the 8×8 MIMO link condition compared to the 4×4 MIMO link.
[0112] As shown in Figure 3, when the RIS coordinates are fixed at (50,5,5), the achievable speed of the system varies with the number of BD-RIS units from M=10 to M=100 under a 4×4 MIMO channel. The BD-RIS speed using the RALM algorithm is much higher than that of the traditional RIS.
[0113] like Figure 4 As shown, the first scenario employs a 4×4 MIMO channel and a BD-RIS with M=100 units. The spectral efficiency (in b / s / Hz) of the system when the BD-RIS moves along the x-axis from xRIS=10 m to xRIS=100 m is illustrated. For each RIS position, the results of 100 independent channel implementations are averaged in this embodiment. The BD-RIS algorithm proposed in this embodiment achieves the highest achievable rate, consistently outperforming the phase-shift optimized diagonal RIS, and both are significantly better than the diagonal RIS using random phase shifts. As a benchmark, Figure 4 also shows the rate without RIS.
[0114] The above are merely embodiments of the present invention and are not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, or improvements made within the spirit and principle of the present invention should be included within the scope of the claims of the present invention.
Claims
1. A scattering matrix optimization method for maximizing the capacity of a diagonally reconfigurable smart surface-assisted MIMO, characterized in that, Includes the following steps: Step 1: Establish a model of the super-diagonal reconfigurable smart surface-assisted MIMO system. Step 2: Based on the model in Step 1, establish the capacity maximization problem and propose a mathematical model, namely the original optimization problem (1). Step 3: Use the alternating optimization method to optimize the two optimization objectives of the original optimization problem (1), namely the covariance matrix and the scattering matrix, and perform alternating optimization. Step 4: Optimize the covariance matrix by fixing the scattering matrix and using the water injection method to achieve the optimality of the covariance matrix. Step 5: Optimize the scattering matrix with a fixed covariance matrix, and use a two-level iterative method to solve the double-constraint optimization problem of the scattering matrix. Step 6: After two-level iterative optimization to convergence, perform alternating optimization to convergence.
2. The method according to claim 1, characterized in that, The solution to the convex optimization problem of the scattering matrix in step four is as follows:
3. The method according to claim 1, characterized in that, Step five, the optimization process for the scattering matrix, specifically includes the following steps: Step a: Initialize system parameters and optimize the scattering matrix. Step b: Use the augmented Lagrangian method to incorporate the symmetric constraints into the original optimization problem (1) to obtain the optimization problem (2), and construct the outer iterative optimization until convergence. Step c: Use the Riemann conjugate gradient method to solve the optimization problem (2), construct the inner iterative optimization, until convergence.
4. The method according to claim 1, characterized in that, The original optimization problem (1) in step two is:
5. The method according to claim 3, characterized in that, The formula for optimization problem (2) in step b is: Y is the Lagrange multiplier, ρ is the penalty factor, Θ is the scattering matrix, and N... R This refers to the number of antennas at the receiving end. It is noise power, H eff yes,(•) T This represents the transpose of the original vector or its conjugate.
6. The method according to claim 3, characterized in that, The auxiliary variables in step b include Y and ρ, and their update formulas are as follows: Where k is the number of iterations and η is the penalty factor amplification coefficient.
7. The method according to claim 3, characterized in that, The Riemann conjugate gradient method formula in step c includes:
8. The method according to claim 3, characterized in that, The auxiliary variables in step c include d and α, and their update formula is: in, γ is the Riemann gradient, β is the conjugate coefficient, and γ is the step size reduction factor.
9. The method according to claim 7, characterized in that, Update the conjugate coefficient in the formula and search direction Its update formula is: in, It is a tangent spatial translation.