Large-scale MIMO downlink precoding manifold optimization method
By employing manifold optimization and Riemann conjugate gradient design methods, the precoder design problem under different power constraints in large-scale MIMO systems was solved, achieving faster convergence and lower complexity, and improving the weighted sum rate performance of the system.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-03-28
- Publication Date
- 2026-04-03
AI Technical Summary
In large-scale MIMO systems, how can we design precoders under different power constraints to suppress inter-user interference, improve system throughput, and reduce computational complexity?
The precoding design problem is transformed into an optimization problem on a Riemannian manifold using a manifold optimization method. The problem is solved iteratively using the Riemann conjugate gradient design method, which includes Riemann gradient, recovery, and vector shifting. It is applicable to total power constraints, user power constraints, and per-antenna power constraints.
It improves convergence speed, reduces computational complexity, enables more efficient precoding matrix design, and enhances the weighted sum rate performance of the system.
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Abstract
Description
Technical Field
[0001] This invention belongs to the field of wireless communication technology and relates to a downlink precoding design method for a maximum weighted sum rate massive MIMO system based on manifold optimization. Background Technology
[0002] Massive multiple-input multiple-output (MIMO) is one of the key technologies in fifth-generation (5G) wireless networks [1]. In massive MIMO systems, base stations (BS) equipped with a large number of antennas can serve multiple user terminals on the same time-frequency resources, providing huge potential capacity gain and high energy efficiency. However, serving multiple users at the same time can also lead to serious inter-user interference, thereby reducing spectrum efficiency. In massive MIMO downlink transmission, the correct design of the precoder is particularly important in order to suppress interference and increase system throughput. Due to its low complexity, linear precoders have been widely studied.
[0003] Since the weighted sum-rate (WSR) directly aims to improve system throughput, which is also one of the main goals of communication systems, it has significant practical implications and is widely considered in massive MIMO. Furthermore, different power constraints are encountered during precoding design. Generally, power constraints in massive MIMO systems can be divided into three main categories: total power constraint (TPC), per-user power constraint (PUPC), and per-antenna power constraint (PAPC). Designing precoders under different power constraints typically presents different problems.
[0004] Generally, the design of a WSR-maximizing precoder under the aforementioned power constraints can be formulated as an optimization problem with equality constraints. In recent years, manifold optimization has been extensively studied and successfully applied in many fields, demonstrating significant advantages in handling challenging equality constraints. In mathematics, a manifold is a topological space that locally resembles Euclidean space at each point. By revealing the inherent geometric properties of equality constraints, manifold optimization reshapes the constrained problem in Euclidean space into an unconstrained optimization problem on a manifold. The concepts of traditional optimization methods for solving unconstrained problems are transformed into concepts on a manifold, and traditional optimization methods are transformed into Riemannian methods solved on manifolds. Furthermore, manifold optimization algorithms often exhibit good performance due to their combination of differential geometry, optimization, and numerical analysis. Therefore, manifold optimization provides a highly compatible, low-complexity, and fast-convergence method for designing optimal WSR-maximizing precoders for different power constraints within a unified framework. Summary of the Invention
[0005] Technical Problem: The purpose of this invention is to provide a maximum weighted sum rate large-scale MIMO downlink precoding manifold optimization method based on manifold optimization, which is applicable to total power constraints, user power constraints and per-antenna power constraints, and accelerates the convergence rate and reduces complexity.
[0006] Technical Solution: To achieve the above objectives, the present invention provides a method for optimizing downlink precoding manifolds in large-scale MIMO, comprising the following steps:
[0007] Step 1: Establish the precoding matrix design problem in Euclidean space with weighted sum rate maximization under total power constraints, user power constraints, and per-antenna power constraints, respectively, and transform it into the precoding matrix design problem in manifold space with weighted sum rate maximization.
[0008] Step 2: Solve for the orthogonal projection Riemann gradient, retraction, and vector shift under the total power constraint, user power constraint, and antenna-by-antenna power constraint in the manifold space;
[0009] Step 3: The Riemann conjugate gradient design method is used to iteratively solve the precoding design problem of maximizing the sum rate in the manifold space.
[0010] Step 1 specifically involves:
[0011] Consider a single-cell massive MIMO system, where one cell is equipped with M t A base station with one antenna serves U user terminals simultaneously, and the i-th user equipment is equipped with M antennas. i There are 1 antenna, and users are evenly and randomly distributed in the cell. The user set is represented as... The set of transmitting antennas on the base station side is represented as follows let This represents the signal sent to the i-th terminal, and satisfies... z i Let the vector of independent and identically distributed complex cyclic symmetric Gaussian noise be represented by the distribution as follows: H i P is the complex channel matrix from the base station to the i-th terminal. i This is the corresponding precoding matrix. The received signal y of the i-th user equipment i It can be represented as:
[0012]
[0013] Assume that the i-th user equipment has access to the effective channel H i P i Given the perfect channel state information, consider the worst-case scenario, assuming interference plus noise. It follows a Gaussian distribution, and its covariance matrix can be written as
[0014]
[0015] This represents the expectation. Assume R... i Given user i, the rate of user i can be written as:
[0016]
[0017] The problem of maximizing the maximum weighted sum rate (WSR) can be formulated as a minimization problem as follows:
[0018]
[0019] Where w i It is the weighting factor for user i, F(P1,P2,…,P U ) = 0 is a power constraint, let f(P1,…,P) = 0. U ) represents the objective function in Euclidean space.
[0020] Stack the precoding matrices from different users and redefine the variables as follows.
[0021] P = (P1, P2, ..., P U (54)
[0022] From the perspective of manifold optimization Naturally belongs to a linear manifold Define Riemannian measure as
[0023]
[0024] in and It is the tangent space Tangent vector on, Indicating the real part, after the Riemannian measure, This forms a Riemannian manifold; in fact, P = (P1, P2, ..., P...). U Let be a point on the product manifold, which is defined as follows:
[0025]
[0026] Similarly, The tangent space can be described as
[0027]
[0028] The Riemann measure defined on it can be defined as the following direct sum.
[0029]
[0030] in, and It is the tangent vector in the tangent space of P. Indicates straight and;
[0031] This method is applicable to three cases: Total Power Constraint (TPC), User Power Constraint (PUPC), and Antenna-by-Antenna Power Constraint (PAPC). Using the stacked precoder matrix P, TPC, PUPC, and PAPC can be expressed as follows:
[0032]
[0033] Without loss of generality, assume that power allocation has been completed for PUPC and that the following conditions are met. Furthermore, in practical applications, equal antenna power constraints are typically considered to effectively utilize the power amplifier capacity of each antenna. In this case, F3(P) can be rewritten as...
[0034] let
[0035]
[0036] Let represent the sets of precoders that satisfy TPC, PUPC, and PAPC, respectively, where ⊙ denotes the Hadamard product. and The three Riemannian submanifolds are formed, and the constrained problems under TPC, PUPC, and PAPC can be transformed into unconstrained problems on the three Riemannian submanifolds, respectively.
[0037]
[0038] in, and These are manifolds and The point on the graph. The superscript * indicates the optimal value.
[0039] Step 2 is as follows:
[0040] Total power constraint:
[0041] tangent space It can be written as the direct sum of two orthogonal quantities.
[0042]
[0043] tangent space It can be written as
[0044]
[0045] yes The orthogonal complement space can be written as
[0046]
[0047] Therefore, any tangent vector It can be decomposed into two orthogonal parts
[0048]
[0049] in and They represent ξ respectively P arrive and The orthogonal projection. Defined as follows: for any Orthographic projection It can be represented as
[0050]
[0051] After having Then, the Riemann gradient is derived and defined.
[0052]
[0053]
[0054]
[0055] The Riemann gradient of f(P) is
[0056] gradf(P)=(gradf(P1),gradf(P2),…,gradf(P U (70)
[0057] in
[0058]
[0059] The Riemann gradient is
[0060]
[0061] in
[0062]
[0063] In fact, the constraint tr(P) H P) = P defines a sphere, and its retraction is defined as
[0064]
[0065] in yes A tangent vector on.
[0066] Vector shift It can be defined as arrive orthographic projection
[0067]
[0068] User power constraints:
[0069] tangent space for:
[0070]
[0071] A formal space can be written as
[0072]
[0073] in It is a subset of a block diagonal matrix with dimension U. Given the normal space, we can deduce that from... arrive Orthogonal projection;
[0074] For any Orthographic projection Represented as
[0075]
[0076] [X] i The block diagonal matrix X = (X1, X2, ..., X...) represents the block diagonal matrix X = (X1, X2, ..., X...) N The i-th submatrix in ) is X i The Riemann gradient is as follows: The Riemann gradient can be expressed as
[0077]
[0078] in
[0079]
[0080] Note Each submatrix of forms a sphere, therefore It itself forms a slanted manifold, and its collapse can be defined by scaling. Let
[0081]
[0082] let For any and
[0083]
[0084] A revocation was defined.
[0085] Similarly, Vector shift It can be obtained through orthographic projection;
[0086] Antenna-by-antenna power constraints:
[0087] tangent space It can be written as
[0088]
[0089] A formal space can be written as
[0090]
[0091] in It is a set of diagonal matrices. Similarly, we can derive orthogonal projection from this.
[0092] For any Orthographic projection It can be represented as
[0093]
[0094] Yes Then, the Riemann gradient can be derived.
[0095] The Riemann gradient on is
[0096]
[0097] in
[0098]
[0099] Each column forms a sphere, so This constitutes a group containing M t The standard oblique manifold of a sphere. Like PUPC, the retraction under PAPC can be similarly defined. Let...
[0100]
[0101] For any
[0102]
[0103] This constitutes a recovery.
[0104] Vector shift It can also be done by... orthogonal projection to Obtained from above.
[0105] Step 3 specifically involves:
[0106] Riemann conjugate gradient design method:
[0107] exist and The precoding update formula on is
[0108]
[0109] Where, α k It's the step length. and These are the search directions of user i on the corresponding manifold, respectively. to indicate and For any manifold in the given information, the search direction of the Riemann conjugate gradient method is written as:
[0110]
[0111] in, It is the Fletcher-Reeves parameter.
[0112] let R at the kth iteration i It can be written as
[0113]
[0114] The Euclidean gradient of the i-th user in the k-th iteration. Written as
[0115]
[0116] in Obtained from equation (47), written as
[0117]
[0118] for and Directly by and Obtained; no further calculation required.
[0119] α k Obtained by the Backtracking method, let (k,n) represent the nth inner iteration in the kth outer iteration. The objective function in the (k,n)th iteration can be considered as having a relationship with the step size α. k,n-1 The function is written as:
[0120]
[0121] in Considered as related to α k,n-1 The function, and using and A similar method was used to obtain:
[0122]
[0123] Similarly, for and It can be by and Obtain directly;
[0124] Implementation of the Riemann conjugate gradient design method algorithm:
[0125] Step a): Confirm the selected Riemannian manifold, and randomly generate or obtain the precoding initial value matrix using the RZF method. Set the initial step size α0 > 0, select constants r ∈ (0,1) and c ∈ (0,1), and set k = 1 and n = 1;
[0126] Step b): Calculation
[0127]
[0128]
[0129]
[0130] Step c): Calculate the Riemann gradient
[0131]
[0132] Step d): Obtain the search direction
[0133] If k≠1,
[0134] If k = 1,
[0135] Step e): α k,n ←rα k,n-1 ,α k,0 =α0.
[0136] Step f): Calculate and
[0137]
[0138]
[0139]
[0140] Step g): Calculate φ(α) k,n ), n←n+1
[0141]
[0142] Step h): If Return to step e);
[0143] Step i):
[0144] Step j): If convergence occurs, output If convergence is not achieved, return to step b).
[0145] The norm of the precoding matrix under the total power constraint is a constant, the set of precoding matrices under the total power constraint forms a sphere, and the set of precoding matrices under the total power constraint is a Riemannian submanifold of the vector space.
[0146] The Forbenius norm of each user's sub-precoding matrix under the user power constraint is constant. The set of precoding matrices under the user power constraint forms an oblique manifold. The set of precoding matrices under the user power constraint is a Riemann submanifold of the vector space.
[0147] The norm of each row vector in the precoding matrix under the antenna power constraint is a constant, and the set of precoding matrices under the antenna power constraint forms an oblique manifold, which is a Riemann submanifold of the vector space.
[0148] In step 2, the Riemann gradient under the total power constraint, user power constraint and antenna-by-antenna power constraint is obtained by orthogonally projecting the gradient on Euclidean space onto the corresponding manifold space.
[0149] Riemann gradient, recovery, and vector shifting under total power constraints, user power constraints, and antenna-per-antenna power constraints;
[0150] Recovery under total power constraints, user power constraints, and per-antenna power constraints, the recovery operator calculation includes:
[0151] The total power constraint is obtained by power normalization of the entire precoding matrix.
[0152] The user power constraint is obtained by normalizing the power of each submatrix of the precoding matrix.
[0153] For antenna-by-antenna power constraints, the power is normalized for each row vector of the precoding matrix.
[0154] Vector shifting under total power constraints, user power constraints, and antenna-by-antenna power constraints is obtained by orthogonally projecting vectors in the current tangent space into the next tangent space.
[0155] Step 3 includes:
[0156] Step 3.1: Obtain the manifold space and initialize the precoding matrix;
[0157] Step 3.2: Calculate the current Riemann gradient and set the conjugate gradient direction to the negative direction of the Riemann gradient;
[0158] Step 3.3: Calculate the user channel multiplied by the user's precoding matrix for each user;
[0159] Step 3.4: For each user, calculate the user channel multiplied by the current search direction of that user.
[0160] Step 3.5: Iteratively solve for the search step size, and update the precoding matrix according to the optimal step size;
[0161] Step 3.6: Determine if convergence has occurred. If convergence has occurred, output the current precoding matrix.
[0162] Step 3.7: Calculate the vector shift using the new precoding matrix and step size to obtain the new conjugate gradient direction, then return to step 3.3;
[0163] Step 3.4 is a Riemann conjugate gradient design method for downlink precoding of large-scale MIMO based on manifold optimization with maximum weighted sum rate. Given the precoding matrix, the objective function of maximum weighted sum rate with respect to the precoding matrix is transformed into a function with respect to the step size. For total power constraints and user power constraints, the user channel left multiplied by the user precoding matrix and the user channel left multiplied by the user search direction, which are required to search the step size, only need to be calculated once.
[0164] Beneficial effects: The present invention solves the precoding matrix under total power constraints, user power constraints and antenna-by-antenna power constraints with fast convergence speed and low complexity of each iteration. The Riemann conjugate gradient design method proposed in this invention can design the precoding matrix under total power constraints, user power constraints and antenna-by-antenna power constraints more efficiently. Attached Figure Description
[0165] Figure 1 This is a geometric schematic diagram of the manifold space optimization of the maximum weighted sum rate large-scale MIMO downlink precoding design method based on manifold optimization of the present invention.
[0166] Figure 2 This is a performance comparison chart of the Riemann conjugate gradient design method and other algorithms under total power constraints.
[0167] Figure 3 This is a performance comparison chart of the Riemann conjugate gradient design method under user power constraints and other algorithms.
[0168] Figure 4 The graph shows a performance comparison between the Riemann conjugate gradient design method and other algorithms under antenna-by-antenna power constraints. Detailed Implementation
[0169] The design of downlink precoding manifold optimization for large-scale MIMO includes the following steps:
[0170] (1) Establish the optimization problem of maximizing the weighted sum rate in Euclidean space under total power constraint, user power constraint and antenna-by-antenna power constraint respectively, and transform it into the optimization problem of maximizing the weighted sum rate in manifold space;
[0171] (2) Solve for the Riemann gradient, recovery and vector shift under three power constraints in the manifold space;
[0172] (3) The Riemann conjugate gradient design method is used to iteratively solve the sum rate maximization problem in the manifold space.
[0173] (3.1) Obtain the manifold space and initialize the precoding matrix;
[0174] (3.2) Calculate the current Riemann gradient and set the conjugate gradient direction to the negative direction of the Riemann gradient;
[0175] (3.3) For each user, calculate the user channel multiplied by the user's precoding matrix;
[0176] (3.4) For each user, calculate the user channel multiplied by the current search direction of that user.
[0177] (3.5) Iteratively solve for the search step size and update the precoding matrix according to the optimal step size;
[0178] (3.6) Determine if convergence has occurred. If convergence has occurred, output the current precoding matrix.
[0179] (3.7) Calculate the vector shifting using the new precoding matrix and step size to obtain the new conjugate gradient direction, and return to (3.3);
[0180] Step (3.4) includes:
[0181] Given the precoding matrix, the original objective function for the maximum weighted sum rate of the precoding matrix is transformed into a function of the step size;
[0182] For total power constraints and user power constraints, the user channel left-multiplied by the user precoding matrix and the user channel left-multiplied by the user search direction, which are required to calculate the search step size, only need to be calculated once.
[0183] I. Establishing the maximum weighted sum rate optimization problem on the manifold space
[0184] We are considering a single-cell massive MIMO system, where the cell contains one device equipped with M... t A base station with one antenna serves U user terminals simultaneously, and the i-th user equipment is equipped with M antennas. i There are one antenna. Users are uniformly and randomly distributed throughout the cell. The user set is represented as... The set of transmitting antennas on the base station side is represented as follows let This represents the signal sent to the i-th terminal, and satisfies... z i Let the vector of independent and identically distributed complex cyclic symmetric Gaussian noise be represented by the distribution as follows: H i P is the complex channel matrix from the base station to the i-th terminal. iThis is the corresponding precoding matrix. The received signal y of the i-th user equipment i It can be represented as
[0185]
[0186] For simplicity, we assume that the i-th user equipment has access to the effective channel H. i P i The perfect channel state information is known. Considering the worst case, we assume interference plus noise. It follows a Gaussian distribution, and its covariance matrix can be written as
[0187]
[0188] Assume R i Given user i, the rate of user i can be written as:
[0189]
[0190] This invention focuses on the problem of maximizing the maximum weighted sum rate (WSR), which can be formulated as the following minimization problem:
[0191]
[0192] Where w i It is the weighting factor for user i, F(P1,P2,…,P U ) = 0 is a power constraint. We let f(P1,…,P) = 0. U ) represents the objective function in Euclidean space.
[0193] First, we stack the precoding matrices of different users and redefine the variables as follows:
[0194] P = (P1, P2, ..., P U (101)
[0195] From the perspective of manifold optimization Naturally belongs to a linear manifold We define the Riemannian measure as
[0196]
[0197] in and It is the tangent space The tangent vector on. This indicates taking the real part. With the Riemannian measure, This forms a Riemannian manifold. In fact, P = (P1, P2, ..., P... U Let be a point on the product manifold, which is defined as follows:
[0198]
[0199] Similarly, The tangent space can be described as
[0200]
[0201] The Riemann measure defined on it can be defined as the following direct sum.
[0202]
[0203] in, and It is the tangent vector in the tangent space of P.
[0204] This invention applies to three cases: Total Power Constraint (TPC), User Power Constraint (PUPC), and Per-Antenna Power Constraint (PAPC). Using a stacked precoder matrix P, TPC, PUPC, and PAPC can be represented as follows:
[0205]
[0206] Without loss of generality, we assume that power allocation has been completed for PUPC and that the following conditions are met. Furthermore, in practical applications, equal antenna power constraint (EAPC) is usually considered to effectively utilize the power amplifier capacity of each antenna, and F3(P) can be rewritten as
[0207] let
[0208]
[0209] Let represent the sets of precoders that satisfy TPC, PUPC, and PAPC, respectively. Here, ⊙ represents the Hadamard product. and These constitute three distinct Riemannian submanifolds. The constrained problems under TPC, PUPC, and PAPC can be transformed into unconstrained problems on these three Riemannian submanifolds.
[0210]
[0211] II. Calculation of Riemannian elements on manifold space
[0212] 1) Total power constraint:
[0213] Tangent space It can be written as the direct sum of two orthogonal quantities.
[0214]
[0215] Tangent space It can be written as
[0216]
[0217] yes The orthogonal complement space can be written as
[0218]
[0219] Therefore, any It can be decomposed into two orthogonal parts
[0220]
[0221] in and They represent ξ respectively P arrive and The orthogonal projection. Defined as follows: for any Orthographic projection It can be represented as
[0222]
[0223] After having Next, we derive the Riemann gradient. We define...
[0224]
[0225]
[0226]
[0227] The Riemann gradient on f(P) is
[0228] gradf(P)=(gradf(P1),gradf(P2),…,gradf(P U (117)
[0229] in
[0230]
[0231] The Riemann gradient on is
[0232]
[0233] in
[0234]
[0235] In fact, the constraint tr(P) H P) = P actually defines a sphere, and its retraction can be defined as
[0236]
[0237] Vector shifting in Riemannian submanifolds It can be defined as arrive orthographic projection
[0238]
[0239] 2) User power constraints:
[0240] Tangent space for:
[0241]
[0242] A formal space can be written as
[0243]
[0244] in It is a subset of a block diagonal matrix with dimension U. Having the normal space, we can deduce from... arrive The orthogonal projection.
[0245] For any Orthographic projection It can be represented as
[0246]
[0247] The Riemann gradient is as follows.
[0248] The Riemann gradient on can be expressed as
[0249]
[0250] in
[0251]
[0252] Note Each submatrix of forms a sphere, therefore It itself forms a slanted manifold, and its collapse can be defined by scaling. Let
[0253]
[0254] let For any and
[0255]
[0256] A revocation was defined.
[0257] Similarly, vector shifting It can be obtained through orthographic projection:
[0258]
[0259] 3) Antenna-by-antenna power constraints:
[0260] tangent space It can be written as
[0261]
[0262] A formal space can be written as
[0263]
[0264] in It is a set of diagonal matrices. Similarly, we can derive orthogonal projection from this.
[0265] For any Orthographic projection It can be represented as
[0266]
[0267] Yes Then, the Riemann gradient can be derived.
[0268] The Riemann gradient on is
[0269]
[0270] in
[0271]
[0272] Each column forms a sphere, so This constitutes a group containing M t The standard oblique manifold of a sphere. Like PUPC, the retraction under PAPC can be similarly defined. Let...
[0273]
[0274] For any
[0275]
[0276] This constitutes a recovery.
[0277] It can also be done by... orthogonal projection to Obtained from:
[0278]
[0279] III. Riemann conjugate gradient design method and algorithm implementation
[0280] 1. Riemann conjugate gradient design method
[0281] Let the superscript k denote the k-th outer iteration. and The precoding update formula on is
[0282]
[0283] Where, α k It's the step length. and These represent the search directions of user i on the corresponding manifold. We use... to indicate and For any manifold in the given region, the search direction of the Riemann conjugate gradient method can be written as:
[0284]
[0285] in, It is the Fletcher-Reeves parameter.
[0286] let R at the kth iteration i It can be written as
[0287]
[0288] The Euclidean gradient of the i-th user in the k-th iteration. Written as
[0289]
[0290] in It can be obtained from equation (47), written as
[0291]
[0292] It can be seen that, for and It can be directly from and Obtained; no further calculation required. α k This can be obtained using the Backtracking method. For ease of description, we use (k,n) to represent the nth inner iteration in the kth outer iteration. The objective function in the (k,n)th iteration can be considered as a function relating to the step size α. k,n-1 The function is written as
[0293]
[0294] in Considered as related to α k,n-1 The function, and can be used with and A similar method was used to obtain:
[0295]
[0296] Similarly, for and It can be by and Obtain directly.
[0297] 2. Implementation of the Riemann conjugate gradient design method algorithm
[0298] Step a): Confirm the selected Riemannian manifold, and randomly generate or obtain the precoding initial value matrix using the RZF method. Set the initial step size α0 > 0, select constants r ∈ (0,1) and c ∈ (0,1), and set k = 1 and n = 1;
[0299] Step b): Calculation
[0300]
[0301]
[0302]
[0303] Step c): Calculate the Riemann gradient
[0304]
[0305] Step d): Obtain the search direction
[0306] If k≠1,
[0307] If k = 1,
[0308] Step e): α k,n ←rα k,n-1 ,α k,0 =α0.
[0309] Step f): Calculate and
[0310]
[0311]
[0312]
[0313] Step g): Calculate φ(α) k,n ), n←n+1
[0314]
[0315] Step h): If Return to step e).
[0316] Step i):
[0317] Step j): If convergence occurs, output If convergence is not achieved, return to step b).
[0318] Implementation effect
[0319] To enable those skilled in the art to better understand the present invention, the following embodiment presents and compares the sum rate performance and convergence speed of the precoding transmission using the maximum weighted sum rate large-scale MIMO downlink precoding Riemann conjugate gradient design method based on manifold optimization under a specific system configuration.
[0320] Considering the base station configuration of a large-scale MIMO system, the number of antennas on the base station side is M. t =128, number of users U=20, each user has 2 receiving antennas and 2 receiving data streams, that is Equal power constraints are considered in PUPC constraints. Comparisons are made with the WMMSE method proposed in “SS Christensen, R. Agarwal, E. De Carvalho, and JMCioffi, “Weighted sum-rate maximization using weighted MMSE for MIMO-BC beamforming design,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 4792–4799, Dec. 2008,” and the ZF-based PAPC method proposed in “J. Choi, S. Han, and J. Joung, “Low-complexity multiuser MIMO precoder design under-antenna power constraints,” IEEE Trans. Veh. Technol., vol. 67, no. 9, pp. 9011–9015, Sep. 2018.” Performance comparisons are shown below. Figure 2 , Figure 3 and Figure 4 As shown, the Riemann conjugate gradient method converges faster and has better speed performance. Meanwhile, the WMMSE method under total power constraint has a complexity of O(n). The complexity of the Riemann conjugate gradient (RCG) design method is... Under user power constraints, the WMMSE method has a time complexity of O(n). The complexity of RCG is... The ZF-Based complexity under total power constraints is The complexity of RCG is... Where N out N represents the number of outer iterations. in Let N be the number of inner iterations and N be the number of ZF-Based iterations. It can be seen that the RCG design method has lower complexity.
[0321] In the embodiments provided in this application, it should be understood that the disclosed methods can be implemented in other ways without departing from the spirit and scope of this application. The current embodiments are merely exemplary examples and should not be considered limiting, nor should the specific content given limit the purpose of this application. For example, some features may be omitted or not implemented.
[0322] The technical means disclosed in this invention are not limited to those disclosed in the above embodiments, but also include technical solutions composed of any combination of the above technical features. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of this invention, and these improvements and modifications are also considered within the scope of protection of this invention.
Claims
1. A method for optimizing downlink precoding manifolds in large-scale MIMO, characterized in that, The method includes the following steps: Step 1: Establish the precoding matrix design problem in Euclidean space with weighted sum rate maximization under total power constraints, user power constraints, and per-antenna power constraints, respectively, and transform it into the precoding matrix design problem in manifold space with weighted sum rate maximization. Step 2: Solve for orthogonal projection, Riemann gradient, retraction, and vector shift under total power constraints, user power constraints, and antenna-by-antenna power constraints in manifold space; Step 3: The Riemann conjugate gradient (RCG) design method is used to iteratively solve the precoding design problem of maximizing the sum rate in the manifold space; Riemann conjugate gradient design method: Use superscript Indicates the first In the next outer iteration, and The precoding update formula on is in, It is the first The step size of the next iteration. and They are the first Secondary user The search direction on the corresponding manifold is used to indicate and For any manifold in the given information, the search direction of the Riemann conjugate gradient method is written as: in, It's the Fletcher-Reeves parameter. let , , No. During the next iteration It can be written as No. The user in the first Euclidean gradient in the next iteration. , written as in Obtained from equation (47), written as for and , Directly by and Obtained; no further calculation required. Obtained by the Backtracking method, using Indicates the first The first iteration in the outer iteration The inner iteration, the... The objective function in the next iteration can be considered as being related to the step size. The function is written as: in Considered as related The function, and using and A similar method was used to obtain: Similarly, for and , It can be by and Obtain directly; Implementation of the Riemann conjugate gradient design method algorithm: Step a): Confirm the selected Riemannian manifold, and randomly generate or obtain the precoding initial value matrix using the RZF method. Set the initial step size Select a constant , ,set up ; Step b): Calculation Step c): Calculate the Riemann gradient Step d): Obtain the search direction like , , like , Step e): , Step f): Calculate and : Step g): Calculate , Step h): If Return to step e); Step i): ; Step j): If convergence occurs, output If convergence is not achieved, return to step b).
2. The method for optimizing downlink precoding manifolds in large-scale MIMO according to claim 1, characterized in that: Step 1 specifically involves: Considering a single-cell massive MIMO system, there is one cell equipped with A base station with one antenna, serving simultaneously The user terminal, the first Each user device is equipped with There are 1 antenna, and users are evenly and randomly distributed in the cell. The user set is represented as... The set of transmitting antennas on the base station side is represented as ,let Indicates to the first The signal sent by each terminal, and satisfies , The dimension is The complex vector space, It is a dimension of The identity matrix; Let the vector of independent and identically distributed complex cyclic symmetric Gaussian noise be represented by the distribution as follows: , From the base station to the Complex channel matrix of each terminal, That is the corresponding precoding matrix. The signal received by each user equipment It can be represented as: Assume the first Each user equipment for valid channels Given the perfect channel state information, consider the worst-case scenario, assuming interference plus noise. It follows a Gaussian distribution, and its covariance matrix can be written as This indicates the expectation; assuming For users It is known that the user The rate can be written as . (3) The problem of maximizing the maximum weighted sum and rate can be formulated as a minimization problem as follows: ， (4) in User The weighting factor, It is a power constraint that allows Describe the objective function in Euclidean space ; Stack the precoding matrices from different users and redefine the variables as follows. From the perspective of manifold optimization Naturally belongs to a linear manifold The Riemannian measure is defined as in and It is the tangent space Tangent vector on, Indicating the real part, after the Riemannian measure, This forms a Riemannian manifold; in fact, It is a point on the cumulative manifold, which is defined as follows: Similarly, The tangent space can be described as The Riemann measure defined on it can be defined as the following direct sum. in, and Is The tangent vector in the tangent space, Indicates straight and; This method is applicable to three cases: total power constraint, user power constraint, and per-antenna power constraint. It utilizes stacked precoder matrices. TPC, PUPC, and PAPC can be represented as follows: This represents finding the trace over a matrix; without loss of generality, it is assumed that power allocation has been performed on PUPC and that the following conditions are met. Furthermore, in practical applications, equal antenna power constraints are typically considered to effectively utilize the power amplifier capacity of each antenna. It can be rewritten as ; let Let T, PUPC, and PAPC represent the sets of precoders that satisfy TPC, PUPC, and PAPC, respectively. Represents the Hadamard product. , and The three Riemannian submanifolds are formed, and the constrained problems under TPC, PUPC, and PAPC can be transformed into unconstrained problems on the three Riemannian submanifolds, respectively. in, , and These are manifolds , and The dot above; superscript This represents the optimal value.
3. The method for optimizing downlink precoding manifolds in large-scale MIMO according to claim 1, characterized in that: Step 2 is as follows: Total power constraint: tangent space It can be written as the direct sum of two mutually orthogonal spaces. tangent space It can be written as yes The orthogonal complement space can be written as Represents real numbers; therefore, any tangent vector It can be decomposed into two orthogonal parts ， (16) in and They represent arrive and Orthogonal projection; Defined as follows For any orthographic projection It can be represented as After having Then, the Riemann gradient is defined as follows: definition The Riemann gradient is in The Riemann gradient is ， (23) in In fact, constraints A sphere is defined, and its retraction is defined as... in yes A tangent vector on; Vector shift , , can be defined as arrive orthographic projection User power constraints: tangent space for: A formal space can be written as ， (28) in It is a subset of a block diagonal matrix with dimension . , express A block diagonal matrix on the diagonal; having a normal space, derive from... arrive Orthogonal projection; For any orthographic projection Represented as Represents a block diagonal matrix The first in Submatrices, i.e. The Riemann gradient is as follows: The Riemann gradient can be expressed as in Note Each submatrix of forms a sphere, therefore It itself forms a slanted manifold, and its retraction can be defined by scaling; let let , For any and , A revocation was defined; Similarly, Vector shift , , It can be obtained through orthographic projection Antenna-by-antenna power constraints: tangent space It can be written as A formal space can be written as in It is a set of diagonal matrices. It is a dimension of The real vector space, Representing vectors A diagonal matrix with elements on the diagonal; similarly, we can derive orthographic projection from this. : For any orthographic projection It can be represented as Yes Then, the Riemann gradient can be derived; The Riemann gradient is in Each column forms a sphere, so This constitutes a collection The standard oblique manifold of a sphere; like PUPC, the retraction under PAPC can be similarly defined; let For any , , This constitutes a recovery; Vector shift , Alternatively, it can be done by... orthogonal projection to above (42)。 4. The method for optimizing downlink precoding manifolds in large-scale MIMO according to claim 1, 2, or 3, characterized in that: The norm of the precoding matrix under the total power constraint is a constant, the set of precoding matrices under the total power constraint forms a sphere, and the set of precoding matrices under the total power constraint is a Riemannian submanifold of the vector space.
5. The method for optimizing downlink precoding manifolds in large-scale MIMO according to claim 1, 2, or 3, characterized in that: The Forbenius norm of each user's sub-precoding matrix under the user power constraint is constant. The set of precoding matrices under the user power constraint forms an oblique manifold. The set of precoding matrices under the user power constraint is a Riemann submanifold of the vector space.
6. The method for optimizing downlink precoding manifolds in large-scale MIMO according to claim 1, 2, or 3, characterized in that: The norm of each row vector in the precoding matrix under the antenna power constraint is a constant, and the set of precoding matrices under the antenna power constraint forms an oblique manifold, which is a Riemann submanifold of the vector space.
7. The method for optimizing downlink precoding manifolds in large-scale MIMO according to claim 1, 2, or 3, characterized in that: In step 2, the Riemann gradient under the total power constraint, user power constraint and antenna-by-antenna power constraint is obtained by orthogonally projecting the gradient on Euclidean space onto the corresponding manifold space. Orthogonal projection, Riemann gradient, retraction, and vector shifting under total power constraints, user power constraints, and antenna-by-antenna power constraints; Recovery under total power constraints, user power constraints, and per-antenna power constraints, the recovery operator calculation includes: The total power constraint is obtained by power normalization of the entire precoding matrix. The user power constraint is obtained by normalizing the power of each submatrix of the precoding matrix. For antenna-by-antenna power constraints, the power is normalized for each row vector of the precoding matrix. Vector shifting under total power constraints, user power constraints, and antenna-by-antenna power constraints is obtained by orthogonally projecting vectors in the current tangent space into the next tangent space.
8. The method for optimizing downlink precoding manifolds in large-scale MIMO according to claim 1, characterized in that: Step 3 includes: Step 3.1: Obtain the manifold space and initialize the precoding matrix; Step 3.2: Calculate the current Riemann gradient and set the conjugate gradient direction to the negative direction of the Riemann gradient; Step 3.3: Calculate the user channel multiplied by the user's precoding matrix for each user; Step 3.4: For each user, calculate the user channel multiplied by the current search direction of that user. Step 3.5: Iteratively solve for the search step size, and update the precoding matrix according to the optimal step size; Step 3.6: Determine if convergence has occurred. If convergence has occurred, output the current precoding matrix. Step 3.7: Calculate the vector shift using the new precoding matrix and step size to obtain the new conjugate gradient direction, and return to step 3.
3.
9. The method for optimizing downlink precoding manifolds in large-scale MIMO according to claim 8, characterized in that: Step 3.4 is a Riemann conjugate gradient design method for downlink precoding of large-scale MIMO based on manifold optimization with maximum weighted sum rate. Given the precoding matrix, the objective function of maximum weighted sum rate with respect to the precoding matrix is transformed into a function with respect to the step size. For total power constraints and user power constraints, the user channel left multiplied by the user precoding matrix and the user channel left multiplied by the user search direction, which are required to search the step size, only need to be calculated once.