Manifold optimization emission pattern synthesis method for analog-digital hybrid array

Abstract

The invention discloses a manifold optimization transmitting directional diagram synthesis method for an analog digital hybrid array. According to the manifold optimization transmitting directional diagram synthesis method for the analog digital hybrid array, the HAD array transmitting directional diagram synthesis problem with the criterion of minimizing the matching error between an expected beam directional diagram and an actual beam directional diagram is researched. According to the method, the solution is carried out by utilizing block coordinate descent (BCD) and Riemannian manifold optimization. Firstly, a constant modulus auxiliary variable is introduced to equivalently represent that an objective function is a smooth biquadratic function; then, alternately optimizing a scaling factor, a digital weight, a simulation weight and a constant modulus auxiliary variable under a BCD framework, reexpressing the simulation weight and the auxiliary variable as a quadratic programming problem under constant modulus constraint, and solving through a Riemannian gradient descent method; the globally optimal solution of the digital weight is obtained through a Lagrange multiplier method. A numerical result shows that compared with other representative algorithms, the algorithm provided by the invention has better beam matching performance.

Description

Technical Field

[0001] The invention belongs to the technical field of array signal processing, and in particular relates to a manifold optimization emission pattern synthesis method for a hybrid analog digital (HAD) array. Background Art

[0002] In radar and millimeter-wave communication applications, large-scale transmit antenna arrays are widely used to enhance target reception and compensate for the high path loss experienced by millimeter-wave signals. However, the use of fully digital (FD) architectures, with each antenna equipped with a separate radio frequency (RF) link, results in prohibitive hardware cost and complexity. As an alternative, the hybrid analog-digital (HAD) architecture offers a promising solution due to its ability to provide flexible connectivity between the RF link and antenna elements using phase shifters. Hybrid analog-digital arrays offer a promising approach to implementing large-scale transmit arrays due to their trade-off between system performance and hardware complexity. Generally, HAD arrays can achieve performance close to that of FD arrays at a lower hardware cost by integrating low-dimensional digital beamformers with high-dimensional analog beamforming. Specifically, they utilize a small number of RF links for digital beamforming and a large number of low-cost analog phase shifters to connect the antenna elements and the RF link for analog beamforming. Based on the connectivity structure between the antenna elements and the RF link, HAD arrays can be broadly categorized into three types: fully connected, partially connected, and dynamically connected. In fully connected and dynamically connected architectures, each RF link must be connected to all antenna elements, either directly or dynamically via analog phase shifters and RF adders. However, multi-input RF adders are expensive in practice. Therefore, a partially connected architecture, where each RF link is connected to a subset of antenna elements, is a more cost-effective HAD architecture.

[0003] Unlike traditional FD arrays, HAD array transmit pattern synthesis is often a difficult problem to solve because all analog weights in the array are subject to constant modulus constraints and the analog and digital weights are heavily coupled in the objective function. Therefore, it is necessary to develop a customized method to solve this problem. Summary of the Invention

[0004] The purpose of the present invention is to provide a manifold optimization transmission pattern synthesis method for analog digital hybrid array.

[0005] The technical solution to the problem of the present invention is: a manifold optimization transmission pattern synthesis method for analog digital hybrid array, the specific implementation steps are as follows:

[0006] Step 1: For the HAD array transmission pattern synthesis problem, the criterion is to minimize the matching error between the expected beam pattern and the actual beam pattern. The optimization model is:

[0007]

[0008] Where p d (θ) is the desired pattern, p(θ) is the actual pattern, Θ is the angle range, and its value is [-π / 2,π / 2]. The pre-weighting factor at angle θ is w(θ), and the scaling factor is α. The number of array elements in the HAD array is MN, the number of array elements in each analog subarray is N, and the number of digital subarrays is M. is the simulated weight of the HAD array, is the numerical weight, P HAD is the total energy of the digital weights;

[0009] Step 2: For the integral operation contained in the objective function of the optimization problem in step 1, discretize the angle θ, convert the integral operation into discrete summation, and divide the angle region Θ into K parts on average. Then the optimization problem in step 1 is equivalently expressed as:

[0010]

[0011] Step 3: Introduce constant modulus auxiliary variables Remove the non-convex modulo operation in the objective function of the optimization problem in step 2; convert the discrete summation form into a vector form, and the optimization problem model in step 2 is further transformed into the following form:

[0012]

[0013] Step 4: Using the BCD framework, decompose the optimization problem model in step 3 into several easily solvable sub-problems;

[0014] Step 5: Solve each sub-problem in step 4 to obtain the optimal scaling factor, digital weight, analog weight and constant modulus auxiliary variable, so as to achieve the HAD array transmission pattern synthesis problem with minimized pattern matching error.

[0015] Furthermore, in step 1, the far-field narrowband signal of the HAD uniform linear array with a partially connected structure is considered, where each simulated subarray is composed of N isotropic unit antennas, the unit spacing between the unit antennas is d, and the steering vector of each simulated subarray at the angle θ is expressed as

[0016]

[0017] Where λ represents the carrier wavelength; each simulated subarray is connected to a radio frequency link, and the steering vector of the M simulated subarrays above constitutes the HAD array is expressed as

[0018]

[0019] Then, the total steering vector of the HAD array is expressed as

[0020]

[0021] Define the weight value of the nth element on the mth simulation sub-array as x (m-1)N+n , m=1,2,...,M,n=1,2,...,N, then the simulation weight of HAD array is expressed as

[0022]

[0023] Define the RF link weight corresponding to the mth simulation subarray as g m , then the digital weight of the HAD array is expressed as

[0024]

[0025] Therefore, the mixing weight of the HAD array is equivalently expressed as

[0026]

[0027] The HAD array transmission pattern is:

[0028]

[0029] Furthermore, in step 3, in the objective function, the discrete summation form is equivalently converted into a vector form, and the optimization problem in step 2 is equivalently converted into:

[0030]

[0031] in, Represents the weighted desired direction map in vector form. Similarly, Represents the matrixed steering vector; introduces constant modulus auxiliary variables |v k |=1,k=1,…,K, further transform the above problem model into the problem model in step 3.

[0032] Furthermore, in step 4, the BCD method is used to decompose the optimization problem in step 3 into several easily solvable sub-problems for iterative solution, which are specifically:

[0033]

[0034] Where, all superscript t+1 represents the result of the t+1th iteration, and superscript t represents the result of the tth iteration.

[0035] Furthermore, the specific process of solving each sub-problem in step 4 to obtain the optimal analog weights, auxiliary variables, scaling factors and digital weights is as follows:

[0036] Step 5.1 The equivalent expression is Where, The Riemann gradient descent method is used to solve the equivalent transformed subproblem and obtain the simulation weights and auxiliary variables of the t+1th iteration;

[0037] Step 5.2 Solution Get the scaling factor for the t+1th iteration;

[0038] Step 5.3 Solution Get the digital weight of the t+1th iteration;

[0039] Step 5.4 determines whether the iteration termination condition is met, that is, whether |G(g t+1 ,v t+1 ,x t+1 ,α t+1 )-G(g t ,v t ,x t ,α t )|≤ξ2, ο2 is the termination tolerance of the algorithm. If it is satisfied, the loop stops and the analog weights and digital weights obtained in the current iteration are output as the optimal analog weights and digital weights. Otherwise, return to step 5.1.

[0040] Furthermore, the Riemann gradient descent method is used to solve the equivalent transformed subproblem. The specific process of obtaining the simulation weights and auxiliary variables for the t+1th iteration is as follows:

[0041] Subproblem (12) is equivalently transformed into the following problem:

[0042]

[0043] Where, To ensure that the objective function in (15) is a positive definite quadratic form, the objective function is loaded diagonally, that is,

[0044]

[0045] Where λ is the diagonal loading value; According to (16), problem (15) is equivalently expressed as

[0046]

[0047] Where Ψ=B H B+λI L ; For the constant modulus constraint of variable y, consider the following complex circular manifold

[0048]

[0049] The tangent space of the manifold at point y is defined as According to (18), problem (17) can be expressed equivalently as

[0050]

[0051] Then, the first-order RMO technique Riemannian gradient descent (RGD) is used to solve it.

[0052] Furthermore, the basic process of using RGD technology to solve the equivalent transformed sub-problem is as follows:

[0053] Step 5.1.1: Obtain the objective function f(y j ) has a Riemann gradient of

[0054]

[0055] in, It is a tangent space Orthogonal projection on , diag(·) represents the operation of diagonalizing a vector into a diagonal matrix;

[0056] Step 5.1.2: Use backtracking line search to adaptively update the step size. The specific method is:

[0057] Initialize step size κ0 = 1, using formula k i+1 =γk i Iteration step, γ is the step coefficient, when the backtracking condition is met, the iteration is stopped and the current iteration is κ i+1 As the final step of the jth round, otherwise i=i+1; the backtracking condition is

[0058]

[0059] Where σ represents the descent parameter, It means that each element of the vector w is projected onto the unit circle, that is,

[0060] Step 5.1.3: Solve the simulation weight for the j+1th iteration using the following formula:

[0061]

[0062] Determine whether the iteration termination condition is met, that is, whether |f(y j+1 )-f(y j )|<ι2, ι2 is the termination tolerance of the algorithm. If it is met, the loop is stopped and the current one is used as the optimal simulation weight for the t+1th round; otherwise, return to step 5.1.1.

[0063] Furthermore, each subproblem in step 4 is solved, and the transformed scaling factor subproblem is solved through a closed-form solution. The specific process of obtaining the scaling factor for the t+1th iteration is:

[0064] Solve subproblem (13) by the following formula

[0065]

[0066] Furthermore, the sub-problems in step 4 are solved, and the transformed digital weight sub-problems are solved through closed-form solutions to obtain the specific process of the digital weights of the t+1th iteration:

[0067] In the HAD array,

[0068]

[0069] in,

[0070]

[0071] Therefore, subproblem (14) is equivalently expressed as

[0072]

[0073] Where d = α t+1 Zv t+1 , A g =E H (X t+1 ) H A,

[0074] The Lagrange multiplier method is used to solve problem (26); the Lagrange function of (26) is defined as

[0075]

[0076] Where ∈ is the Lagrange multiplier; according to the first-order necessary conditions of problem (26), we can get

[0077]

[0078] definition The characteristic decomposition of

[0079]

[0080] Where T represents the eigenvector matrix, Γ=Diag([γ1,...,γ M ]) represents the diagonal matrix of eigenvalues, and γ1≥γ2≥...≥γ M ; Using (33), equation (28) can be expressed equivalently as

[0081]

[0082] in, definition Simplifying equation (30) to

[0083]

[0084] When ∈≥-γ M When ∈≥-γ M It is monotonically decreasing when ; Equation (31) can be solved by line search to obtain ∈ * . Then, we can get g t+1 The closed-form solution is

[0085]

[0086] Compared with the existing technology, the present invention has the following significant advantages: (1) Since the simulation weights and constant modulus auxiliary variables are updated simultaneously, the proposed algorithm reduces one iteration term and improves the convergence speed. (2) Due to the superior performance of Riemannian manifold optimization, the proposed algorithm also has superior performance in aspects such as transmission pattern synthesis compared with other representative methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0087] Figure 1 is the multi-mainlobe focused beam pattern.

[0088] Figure 2 It is a double main lobe flat-top beam pattern.

[0089] Figure 3 is the cosecant square beam pattern. DETAILED DESCRIPTION

[0090] Based on block coordinate descent (BCD) and first-order Riemannian manifold optimization (RMO) technology, the present invention proposes a HAD array transmit pattern synthesis method based on the BCD framework. This method is applicable to the HAD array transmit pattern synthesis problem with the criterion of minimizing the matching error between the expected beam pattern and the actual beam pattern. Within the BCD framework, the algorithm reconstructs the analog weight and auxiliary variable subproblems into a uni-modular constrained quadratic programming (UCQP) problem, and uses the first-order RMO method to effectively solve the subproblem. The Lagrange multiplier method is used to provide the global optimal solution of the digital weights under energy constraints.

[0091] The technical solution adopted by the present invention is to adopt the concept of BCD and utilize the first-order Riemannian manifold optimization (RMO) technique, namely the Riemannian gradient descent (RGD) method, to propose an efficient algorithm for the Riemannian manifold optimization transmission pattern synthesis problem of analog-digital hybrid arrays within the BCD framework. The basic concept of the algorithm is to convert a non-smooth objective function into a smooth objective function, and then alternately optimize the scaling factor, digital weights, and analog weights. In the algorithm, a constant modulus auxiliary variable is introduced to eliminate the modulo operation in the objective function, thereby equivalently expressing the objective function as a smooth biquadratic function. Then, according to the BCD principle, the scaling factor, digital weights, analog weights, and constant modulus auxiliary variable are alternately updated. The scaling factor and digital weights each have a closed-form solution, and the analog weights and constant modulus auxiliary variable are simultaneously updated via Riemannian gradient descent.

[0092] The following is a further description with reference to the accompanying drawings and embodiments.

[0093] The present invention is a method for synthesizing the transmission pattern of manifold optimization for analog digital hybrid arrays. This method adopts the idea of block coordinate descent and utilizes the first-order Riemannian manifold optimization technology - the Riemannian gradient descent method, to propose an efficient algorithm for the synthesis problem of Riemannian manifold optimization transmission pattern for analog digital hybrid arrays based on the BCD framework. The basic idea of the algorithm is to transform the non-smooth objective function into a smooth objective function, and then alternately optimize the scaling factor, digital weights and analog weights. First, a constant modulus auxiliary variable is introduced to eliminate the modulo operation in the objective function, thereby equivalently expressing the objective function as a smooth biquadratic function. Then, according to the BCD principle, the scaling factor, digital weights, analog weights and constant modulus auxiliary variables are alternately updated, wherein the scaling factor and digital weights have a closed-form solution, and the analog weights and constant modulus auxiliary variables are simultaneously updated through Riemannian gradient descent. The specific implementation steps of this method are as follows:

[0094] Step 1: Consider the far-field narrowband signal of a HAD uniform linear array with a partially connected structure, where each simulated subarray consists of N isotropic unit antennas with a unit spacing of d. The steering vector of each simulated subarray at angle θ can be expressed as

[0095]

[0096] Where λ represents the carrier wavelength. Each simulated subarray is connected to a radio frequency link. The steering vector of the HAD array composed of M simulated subarrays can be expressed as

[0097]

[0098] Therefore, the total steering vector of the HAD array can be expressed as

[0099]

[0100] If the weight value of the nth element on the mth simulation sub-matrix is defined as x (m-1)N+n , m=1,2,...,M,n=1,2,...,N, then the simulation weight of the HAD array can be expressed as

[0101]

[0102] Define the RF link weight corresponding to the mth simulation subarray as g m , then the digital weight of the HAD array can be expressed as

[0103]

[0104] Therefore, the mixing weight of the HAD array can be equivalently expressed as

[0105]

[0106] The emission pattern of the HAD array is

[0107]

[0108] For the synthesis of the transmission pattern of the HAD array, the criterion is generally to minimize the matching error between the expected beam pattern and the actual beam pattern. The expected pattern is defined as p d (θ), the actual pattern is p(θ), the angle range Θ is [-π / 2,π / 2], the pre-weighting factor at angle θ is w(θ), the scaling factor is α, and the simulated weight of the HAD array is The numerical weight is The total energy of the digital weight is P HAD Taking minimizing the matching error between the desired beam pattern and the actual beam pattern as the criterion, the optimization model is:

[0109]

[0110] Step 2: For the integral operation contained in the objective function in problem (40), discretize the angle and convert the integral operation into a discrete summation. Divide the angle region Θ into K equal parts, and then problem (40) can be equivalently expressed as:

[0111]

[0112] Step 3: Introduce the constant modulus auxiliary variable v to remove the non-convex modulus operation in the objective function. First, convert the discrete summation form into a vector form, and the problem (41) is further transformed into the following form:

[0113]

[0114] in, represents the desired direction in vector form, Then, introduce the constant modulus auxiliary variable |v k |=1,k=1,…,K, remove the modulo operation in the objective function and transform the problem (42) into

[0115]

[0116] in, Obviously, when When , Problem (43) has the same solution as Problem (42).

[0117] Step 4: Use a method based on the BCD framework to decompose the non-convex problem into several easily solvable sub-problems, as follows:

[0118]

[0119] Convert problem (43) into solving subproblems, and.

[0120] Step 5: Use the BCD method to decompose problem (43) into several easily solvable subproblems and solve them iteratively. Among them, the subproblems with constant modulus constraints are merged and updated, and transformed into the RMO problem on the complex circular manifold. According to the BCD principle, the optimal solution of {α, g, x, v} is found through the following iterative process. Solve each subproblem in step 4 to obtain the optimal analog weights and digital weights. The specific process is as follows:

[0121] Step 5.1 The equivalent expression is Where, The Riemann gradient descent method is used to solve the equivalent transformed subproblem and obtain the simulation weights and auxiliary variables of the t+1th iteration. The specific process is as follows:

[0122] For the subproblem, its objective function can be equivalently expressed as

[0123]

[0124] in, definition And the combined {x,v} is expressed as

[0125]

[0126] Where L = K + MN. Then, the subproblem is equivalently transformed into the following problem:

[0127]

[0128] In order to ensure that the objective function is a positive definite quadratic form, the diagonal loading technique is used for loading, that is,

[0129]

[0130] Where λ is the diagonal loading value. According to , the problem can be equivalently expressed as

[0131]

[0132] Where Ψ=B H B+λI L For the constant modulus constraint of variable y, consider the following complex circular manifold

[0133]

[0134] The tangent space of the manifold at point y is defined as According to the formula, the problem can be expressed as

[0135]

[0136] Then use RGD to solve it.

[0137] The basic process of solving problem (53) using RGD technology is as follows:

[0138] Step 5.1.1: Obtain the objective function f(y j ) has a Riemann gradient of

[0139]

[0140] in, It is a tangent space Orthogonal projection on , diag(·) represents the operation of diagonalizing a vector into a diagonal matrix.

[0141] Step 5.1.2: Use backtracking line search to adaptively update the step size. The specific method is:

[0142] Initialize step size κ0 = 1, use formula κ i+1 =γκ i Iteration step, γ is the step coefficient, when the backtracking condition is met, the iteration is stopped and the current iteration is κ i+1 As the final step of the jth round, otherwise i=i+1.

[0143] The specific backtracking conditions are:

[0144]

[0145] Where σ represents the descent parameter, It means that each element of the vector w is projected onto the unit circle, that is, Step 5.1.3: Solve the simulation weight for the j+1th iteration using the following formula:

[0146]

[0147] Determine whether the iteration termination condition is met, that is, whether |f(y j+1 )-f(y j )|<ι2, ι2 is the termination tolerance of the algorithm. If it is met, the loop is stopped and the current one is used as the optimal simulation weight for the t+1th round; otherwise, return to step 5.1.1.

[0148] Step 5.2 Solution The specific process is:

[0149] For this subproblem, obtain the scaling factor α for the t+1th iteration t+1 The closed-form optimal solution is

[0150]

[0151] Step 5.3 Solution The specific process is:

[0152] For this subproblem, ignoring the variables that are not related to g in the iteration, the subproblem for g can be described as

[0153]

[0154] In the HAD array,

[0155]

[0156] in,

[0157]

[0158] Therefore, subproblem (58) can be equivalently expressed as

[0159]

[0160] Where d = α t+1 Zv t+1 , A g =E H (X t+1 ) H A,

[0161] Problem (61) can be solved using the Lagrange multiplier method. The Lagrange function of (61) is defined as

[0162]

[0163] Where ∈ is the Lagrange multiplier. According to the first-order necessary conditions of problem (61), we can get

[0164]

[0165] definition The characteristic decomposition of Where T represents the eigenvector matrix, Γ=Diag([γ1,...,γ M ]) represents the diagonal matrix of eigenvalues, and γ1≥γ2≥...≥γ M Therefore, equation (63) can be expressed equivalently as

[0166]

[0167] in, definition Simplifying equation (64) to

[0168]

[0169] When ∈≥-γ M When , the left side of equation (65) is monotonically decreasing. Equation (65) can be solved by line search to obtain ∈ * . Then, the closed-form solution of g can be obtained as

[0170]

[0171] Step 5.4 determines whether the iteration termination condition is met, that is, whether |G(g t+1 ,v t+1 ,x t+1 ,α t+1 )-G(g t ,v t ,x t ,α t )|≤ξ2, ξ2 is the termination tolerance of the algorithm. If it is satisfied, the loop stops and the analog weights and digital weights obtained in the current iteration are output as the optimal analog weights and digital weights. Otherwise, return to step 5.1.

[0172] Example

[0173] The Matlab simulation is used to further illustrate the Riemannian manifold optimization transmission pattern synthesis method for simulating a digital hybrid array.

[0174] 1) Simulation system parameter settings

[0175] Unless otherwise specified, the HAD array used in each simulation has N=6, M=8, all elements are evenly distributed, and the element spacing is d=λ / 2. The spatial angular domain [-90°, 90°] is divided into 181 discrete grids with an interval of 1°, and ξ1=ξ2=10 -5 and T max =J max = 5000. The array mixing weight variables and constant modulus auxiliary variables are randomly initialized. In addition, the fully digital array (FD) and the fully analog array (FA) are selected as references for the HAD array.

[0176] 2) Beam pattern drawing

[0177] To visually demonstrate the transmit beamforming effect of the HAD array, this embodiment uses the proposed method to plot the transmit beam pattern and compares it with two typical HAD transmit pattern synthesis methods: the alternating direction multiplier method (ADMM) and the two-stage method. The horizontal axis of the beam pattern represents the angular range of [-90°, 90°], and the vertical axis is in dB.

[0178] 3) Metrics

[0179] In the present invention, the effect of the final beamforming needs to be measured. In addition to plotting a beam pattern to show the difference from the desired beam pattern, the peak sidelobe level (PSL) is used to quantitatively measure the sidelobe level of the transmit beamforming. The definition of PSL is:

[0180]

[0181] Where: Θ s is the side lobe area, P(θ)=|a H (θ)w| is the level formula. Under the same initial conditions, the smaller the PSL, the better the radar and communication system's transmit beamforming performance in suppressing side lobes.

[0182] 4) Result analysis

[0183] The present invention conducts three example simulations in total, covering focused beams, flat-top beams, and cosecant squared beams commonly used in radar and communication systems. Figure 1 is a multi-mainlobe focused beam pattern, Figure 2 It is a multi-mainlobe flat-top beam. Figure 3 is the cosecant square beam. Where, Figure 3 In the case of a HAD array with N=12 and M=4, a cosecant square beam pattern for sidelobe control is adopted.

[0184] pass Figure 1 , it can be seen that the mainlobe gains of the three HAD array algorithms are comparable. In the three beams on the right, the algorithm is slightly better than the other two algorithms. From the perspective of the [58°, 62°] amplified beam, the mainlobe gain of the proposed method reaches 26.63dB, while the mainlobe gains of ADMM and Two-stage are only 26.32dB and 25.68dB, respectively. On the other hand, the mainlobe gains of the FA and FD arrays are 25.98dB and 26.93dB, respectively. In addition, the PSL of the proposed method is 16.69dB, which is only 2.41dB higher than that of the FD array. Therefore, the HAD array makes a good compromise between beam matching performance and hardware complexity.

[0185] pass Figure 2 , it can be seen that under the HAD array, the proposed method produces the lowest PSL and almost the highest mainlobe ripple compared to the other two algorithms, resulting in the optimal ratio of PSL to mainlobe ripple. At the same time, the PSL of Two-stage and ADMM are 3.09dB and 3.60dB higher than that of the proposed method, respectively. In addition, compared with the FD array, the proposed method has almost no gain loss in the mainlobe region, with only a PSL increase of 2.37dB. In radar applications, this loss is acceptable because mainlobe gain is one of the most important indicators for long-range target detection.

[0186] pass Figure 3 , it can be seen that only ADMM, the proposed method, and the FD array successfully synthesized a cosecant squared beam. Their mainlobe ripple values are 3.89dB, 2.34dB, and 1.62dB, respectively, indicating that the proposed method has better complex mainlobe formation capabilities than ADMM and Two-stage. In addition, the PSLs of ADMM, the proposed method, and the FD array are 14.69dB, 14.19dB, and 13.37dB, respectively. From the perspective of mainlobe ripple and PSL, the proposed method achieves the best beam pattern performance and is closest to the FD array.

[0187] In summary, the method described in this invention exhibits excellent overall performance. Compared with other representative methods, the proposed method achieves near-optimal mainlobe gain performance in commonly used beams and ensures low PSL. Applications in radar and communication systems can significantly save hardware costs and reduce computational complexity with minimal performance loss, thus offering high practical value.

Claims

1. A method for synthesizing transmission patterns using manifold optimization for analog-digital hybrid arrays, characterized by: The specific implementation steps are as follows: Step 1: For the HAD array transmission pattern synthesis problem, the criterion is to minimize the matching error between the expected beam pattern and the actual beam pattern. The optimization model is: Where p d (θ) is the desired pattern, p(θ) is the actual pattern, Θ is the angle range, and its value is [-π / 2,π / 2]. The pre-weighting factor at angle θ is w(θ), and the scaling factor is α. The number of array elements in the HAD array is MN, the number of array elements in each analog subarray is N, and the number of digital subarrays is M. is the simulated weight of the HAD array, is the numerical weight, P HAD is the total energy of the digital weights; Step 2: For the integral operation contained in the objective function of the optimization problem in step 1, discretize the angle θ, convert the integral operation into discrete summation, and divide the angle region Θ into K parts on average. Then the optimization problem in step 1 is equivalently expressed as: Step 3: Introduce constant modulus auxiliary variables Remove the non-convex modulo operation in the objective function of the optimization problem in step 2; convert the discrete summation form into a vector form, and the optimization problem model in step 2 is further transformed into the following form: Step 4: Using the BCD framework, decompose the optimization problem model in step 3 into several easily solvable sub-problems; Step 5: Solve each sub-problem in step 4 to obtain the optimal scaling factor, digital weight, analog weight and constant modulus auxiliary variable, so as to achieve the HAD array transmission pattern synthesis problem with minimized pattern matching error.

2. The method for synthesizing a manifold-optimized transmission pattern for an analog-digital hybrid array according to claim 1, wherein: In step 1, the far-field narrowband signal of a HAD uniform linear array with a partially connected structure is considered, where each simulated subarray is composed of N isotropic unit antennas, and the unit spacing between the unit antennas is d. The steering vector of each simulated subarray at an angle θ is expressed as Where λ represents the carrier wavelength; each simulated subarray is connected to a radio frequency link, and the steering vector of the M simulated subarrays above constitutes the HAD array is expressed as Then, the total steering vector of the HAD array is expressed as Define the weight value of the nth element on the mth simulation sub-array as x (m-1)N+n , m=1,2,...,M,n=1,2,...,N, then the simulation weight of HAD array is expressed as Define the RF link weight corresponding to the mth simulation subarray as g m , then the digital weight of the HAD array is expressed as Therefore, the mixing weight of the HAD array is equivalently expressed as The HAD array transmission pattern is:

3. The method for synthesizing a manifold-optimized transmission pattern for an analog-digital hybrid array according to claim 1, wherein: In step 3, in the objective function, the discrete summation form is equivalently converted into a vector form, and the optimization problem in step 2 is equivalently converted into: in, Represents the weighted desired direction map in vector form. Similarly, Represents the matrixed steering vector; introduces constant modulus auxiliary variables |v k |=1,k=1,…,K, further transform the above problem model into the problem model in step 3.

4. The method for synthesizing a manifold-optimized transmission pattern for an analog-digital hybrid array according to claim 1, wherein: In step 4, the BCD method is used to decompose the optimization problem in step 3 into several easily solvable sub-problems for iterative solution, which are specifically: Where, all superscript t+1 represents the result of the t+1th iteration, and superscript t represents the result of the tth iteration.

5. The method for synthesizing a manifold-optimized transmission pattern for an analog-digital hybrid array according to claim 4, wherein: The specific process of solving each sub-problem in step 4 to obtain the optimal simulation weights, auxiliary variables, scaling factors and digital weights is as follows: Step 5.1 The equivalent expression is st|y l |=1,l=1,...,L., where The Riemann gradient descent method is used to solve the equivalent transformed subproblem and obtain the simulation weights and auxiliary variables of the t+1th iteration; Step 5.2 Solution Get the scaling factor for the t+1th iteration; Step 5.3 Solution Get the digital weight of the t+1th iteration; Step 5.4 determines whether the iteration termination condition is met, that is, whether |G(g t+1 ,v t+1 ,x t+1 ,α t+1 )-G(g t ,v t ,x t ,α t )|≤ξ2, ξ2 is the termination tolerance of the algorithm. If it is satisfied, the loop stops and the analog weights and digital weights obtained in the current iteration are output as the optimal analog weights and digital weights. Otherwise, return to step 5.

1.

6. The method for synthesizing a manifold-optimized transmission pattern for an analog-digital hybrid array according to claim 4, wherein: The Riemann gradient descent method is used to solve the equivalent transformed subproblem. The specific process of obtaining the simulation weights and auxiliary variables for the t+1th iteration is as follows: Subproblem (12) is equivalently transformed into the following problem: Where, To ensure that the objective function in (15) is a positive definite quadratic form, the objective function is loaded diagonally, that is, Where λ is the diagonal loading value; According to (16), problem (15) is equivalently expressed as Where Ψ=B H B+λI L ; For the constant modulus constraint of variable y, consider the following complex circular manifold The tangent space of the manifold at point y is defined as According to (18), problem (17) can be expressed equivalently as Then, the first-order RMO technique Riemannian gradient descent RGD is used to solve it.

7. The method for synthesizing a manifold-optimized transmission pattern for an analog-digital hybrid array according to claim 5, wherein: The basic process of using RGD technology to solve the equivalent transformed sub-problem is as follows: Step 5.1.1: Obtain the objective function f(y j ) has a Riemann gradient of in, It is a tangent space Orthogonal projection on , diag(·) represents the operation of diagonalizing a vector into a diagonal matrix; Step 5.1.2: Use backtracking line search to adaptively update the step size. The specific method is: Initialize step size κ0 = 1, using formula k i+1 =γκ i Iteration step, γ is the step coefficient, when the backtracking condition is met, the iteration is stopped and the current iteration is κ i+1 As the final step of the jth round, otherwise i=i+1; the backtracking condition is Where σ represents the descent parameter, It means that each element of the vector w is projected onto the unit circle, that is, Step 5.1.3: Solve the simulation weight for the j+1th iteration using the following formula: Determine whether the iteration termination condition is met, that is, whether |f(y j+1 )-f(y j )|<ι2, l2 is the termination tolerance of the algorithm. If it is met, the loop is stopped and the current one is used as the optimal simulation weight for the t+1th round; otherwise, return to step 5.1.

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8. The method for synthesizing a manifold-optimized transmission pattern for an analog-digital hybrid array according to claim 4, wherein: Solve each subproblem in step 4, solve the transformed scaling factor subproblem through a closed-form solution, and obtain the scaling factor for the t+1th iteration as follows: Solve subproblem (13) by the following formula 9. The method for synthesizing a manifold-optimized transmission pattern for an analog-digital hybrid array according to claim 4, wherein: Solve each subproblem in step 4, solve the transformed digital weight subproblem through closed-form solution, and obtain the specific process of digital weight for the t+1th iteration: In the HAD array, in, Therefore, subproblem (14) is equivalently expressed as where d = α t+1 Zv t+1 , A g = E H (X t+1 ) H A, The Lagrange multiplier method is used to solve problem (26); the Lagrange function of (26) is defined as Where ∈ is the Lagrange multiplier; according to the first-order necessary conditions of problem (26), we can get definition The characteristic decomposition of Where T represents the eigenvector matrix, Γ=Diag([γ1,...,γ M ]) represents the diagonal matrix of eigenvalues, and γ1≥γ2≥...≥γ M ; Using (33), equation (28) can be expressed equivalently as in, definition Simplifying equation (30) to When ∈≥-γ M When ∈≥-γ M It is monotonically decreasing when ; Equation (31) can be solved by line search to obtain ∈ * . Then, we can get g t+1 The closed-form solution is

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