Conjugate gradient beamforming generation method based on manifold optimization
Patent Information
- Application Number
- CN202310933943.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-07-27
- Publication Date
- 2026-10-27
- Estimated Expiration
- 2043-07-27
AI Technical Summary
[0004]本申请提供一种基于流形优化的共轭梯度波束成形生成方法，用以解决恒模约束下的非凸和NP难的问题
[0075]本申请提供的一种无需松弛的直接方法来解决非凸和NP难的问题，与现有方法相比，在干扰方向上能形成更低的零陷,进而获得更高的信干噪比。且运行时间较短。
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Figure CN116887307B_ABST
Abstract
Description
Technical Field
[0001] This application relates to the field of beamforming technology, and in particular to a method for generating conjugate gradient beamforming based on manifold optimization. Background Technology
[0002] Beamforming technology is a key technology in wireless communication and radar systems, with wide applications. By creating wide nulls, interference can be effectively suppressed. Furthermore, power loss issues can be addressed through phase-only constraints and constant mode constraints (CMC). Therefore, beamforming designs with only phase variations under constant mode constraints have attracted particular attention.
[0003] Due to the constant modulus constraint, this problem becomes non-convex and NP-hard, making it difficult to solve directly. Solving the problem by relaxing the constant modulus constraint requires matrix inverse calculations, which incurs significant computational costs in each iteration. Summary of the Invention
[0004] This application provides a conjugate gradient beamforming generation method based on manifold optimization to solve the non-convex and NP-hard problems under constant modulus constraints.
[0005] In a first aspect, this application provides a conjugate gradient beamforming generation method based on manifold optimization, comprising:
[0006] Construct a problem model that minimizes the energy of the interference signal while ensuring the desired energy;
[0007] By transforming the objective function, the problem model of minimizing the interference signal energy is transformed into a problem model on a complex circular manifold. The problem model on the complex circular manifold is solved by gradient descent. The initial waveform value is called to solve for the output waveform of the optimal expected signal energy in the interference direction.
[0008] Furthermore, the construction of the problem model that minimizes the interference signal energy while ensuring the desired energy includes:
[0009] For a multidimensional antenna array, generate an antenna array weighted vector that satisfies the constant modulus constraint;
[0010] Calculate the expected signal energy in the direction of interference after filtering, and construct a problem model that minimizes the interference signal energy while ensuring the expected energy.
[0011] Furthermore, by utilizing the computational properties of the Kronecker product and the Lagrange multiplier method, the problem model for minimizing the interference signal energy is transformed into a problem model on a complex circular manifold, including:
[0012] The constant modulus constraint problem in the problem model of minimizing the energy of the interference signal is transformed into an unconstrained problem model on a complex circular manifold. The conjugate gradient algorithm based on manifold optimization is derived, and the unconstrained problem model on the complex circular manifold is solved using the conjugate gradient algorithm.
[0013] Furthermore, the expected signal energy in the filtered interference direction corresponding to the minimum interference signal energy that satisfies the convergence condition of the conjugate gradient algorithm is obtained. Substituting this into the initial descent direction and initial wavelength, the optimal output waveform in the interference direction is solved.
[0014] Furthermore, by transforming the objective function, the problem of minimizing the interference signal energy is transformed into a problem model on a complex circular manifold. Detailed steps include:
[0015] The steering vector of a one-dimensional M-element antenna array is
[0016]
[0017] Where r m m = 1, 2, 3, ..., M is the distance between the m-th array element and the first array element. The angle representing signal transmission or reception is λ, where λ is the wavelength; w = [w(1), w(2), ... w(M)] T ∈C M×1 The weighted vector for the antenna array;
[0018] Where w(m), m=1,2,…M represents the weighting factor of the m-th array element, and satisfies the constant modulus constraint.
[0019] |w(m)|=1,m=1,2…M (2)
[0020] The desired energy signal in the θ direction after filtering can be expressed as:
[0021] G(θ) = |w H a(θ)| 2 =w H a(θ)a(θ) H w = w H x(θ)w (3)
[0022] in,
[0023] After filtering at θ i The signal energy of the nearby interference signal is
[0024]
[0025] in, θ i Θ is the angle of arrival of the interference signal. i It is θi Given the error angle range centered on the target, and minimizing the interference signal energy while ensuring the desired energy as much as possible, the problem model for minimizing the interference signal energy is as follows:
[0026]
[0027] Where A0 is the energy of the desired signal.
[0028] Furthermore, the process of transforming the objective function to convert the problem model of minimizing the interference signal energy into a problem model on a complex circular manifold, solving the problem model on the complex circular manifold using the gradient descent method, calling the initial waveform values, and solving for the output waveform of the optimal expected signal energy in the interference direction, wherein the process of transforming the objective function to convert the problem model of minimizing the interference signal energy into a problem model on a complex circular manifold includes:
[0029] The problem model of minimizing the energy of the interference signal is transformed into an optimization problem on the manifold Ω:
[0030] Ω={w∈C M ||w(m)|=1,m=1,2,…M} (6)
[0031]
[0032] Using the Lagrange multiplication method, problem (7) can be written as
[0033]
[0034] Where λ is the Lagrange multiplier, λ>0, problem (8) can be further simplified to
[0035]
[0036] in
[0037] The problem in equation (9) is an unconstrained optimization problem on a complex circular manifold.
[0038] Furthermore, for the unconstrained optimization problem on the complex circular manifold in equation (9), it is solved using the conjugate gradient algorithm based on manifold optimization, including:
[0039] Calculate the Riemann gradient;
[0040] Choose the descent direction;
[0041] Find the appropriate step size based on the direction of descent;
[0042] Finally, the data is continuously updated using an iterative method until the convergence condition is met.
[0043] Furthermore, the detailed steps of the conjugate gradient algorithm based on manifold optimization are as follows:
[0044] Step A: Obtain the Riemann gradient:
[0045] The Riemann gradient is obtained by projecting the Euclidean gradient onto the tangent space, and in the i-th iteration, the Riemann gradient is...
[0046]
[0047] in,(·) T w represents the transpose of an element i Let be an arbitrary given point in space Ω, where w i The tangent vector at point Ω is defined as passing through w on Ω. i A vector that is tangent to any smooth curve. It is a tangent space, formed by the midpoint w of the point space Ω. i Composed of all tangent vectors;
[0048]
[0049] Where R(·) is the real part, and ⊙ represents the Hadamard matrix multiplication;
[0050] The projection operator from Riemannian space to the set of tangent points is expressed as follows:
[0051]
[0052] The Euler gradient is represented as follows:
[0053]
[0054] Step B: Select the descent direction η: The descent direction of the Polak-Ribiere conjugate gradient contains second-order information;
[0055] The descent method for the i-th iteration is as follows:
[0056]
[0057] in, The vector transfer operation required to add or subtract points in different tangent spaces is represented as:
[0058]
[0059] β i PR These are the Polak-Ribieres conjugate parameters, expressed as
[0060]
[0061] Step C: Find the step size
[0062] The step size is found using the Armijo line search strategy, and the smallest integer h ≥ 0 that satisfies the following condition:
[0063]
[0064] Where σ,β∈(0,1),τ i-1 >0, the step size of the i-th iteration is
[0065] τ i =τ i-1 β h (18)
[0066] Step D: Update the obtained points: First, in tangent space A provisional point is obtained above, denoted as
[0067]
[0068] Then, substituting the found points back onto the complex manifold circle, it can be represented as:
[0069]
[0070] in, is Hadamard element-wise deviation, It is the modulus of this element.
[0071] Furthermore, the analysis also includes a computational complexity analysis of the conjugate gradient algorithm based on manifold optimization, including the complexity analysis for each iteration as follows:
[0072] Calculating the Riemann gradient requires (M) 2 +2M) multiplication operations; calculating the descent direction requires (5M) multiplication operations and M division operations; calculating the step size requires ((h+1)M 2 +(h+1)M) multiplication operations; updating requires M multiplication operations and M division operations.
[0073] On the other hand, this application provides a computer-readable storage medium, characterized in that the computer-readable storage medium stores computer-executable instructions, which, when executed by a processor, are used to implement the method as described in any one of the first aspects.
[0074] This application has the following advantages and beneficial effects:
[0075] This application provides a direct method without relaxation to solve the non-convex and NP-hard problems. Compared with existing methods, it can form a lower null in the interference direction, thereby achieving a higher signal-to-interference-plus-noise ratio (SNR). Furthermore, it has a shorter runtime. Attached Figure Description
[0076] The accompanying drawings, which are included to provide a further understanding of the embodiments of this application and form part of this application, do not constitute a limitation on the embodiments of this application. In the drawings:
[0077] Figure 1 The objective function convergence curve of the conjugate gradient beamforming generation method based on manifold optimization provided in an example embodiment of this application is shown.
[0078] Figure 2 Comparison of notch depths for a conjugate gradient beamforming generation method based on manifold optimization provided in another example embodiment of this application.
[0079] Figure 3 A comparison of runtime of a manifold optimization-based conjugate gradient beamforming generation method provided in another example embodiment of this application. Detailed Implementation
[0080] Exemplary embodiments will now be described in detail, examples of which are illustrated in the accompanying drawings. When the following description relates to the drawings, unless otherwise indicated, the same numbers in different drawings denote the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with this application. Rather, they are merely examples of apparatuses and methods consistent with some aspects of this application as detailed in the appended claims.
[0081] Beamforming technology is a key technology in wireless communication and radar systems, with wide applications. By creating wide nulls, interference can be effectively suppressed. Furthermore, power loss issues can be addressed through phase-only constraints and constant mode constraints (CMC). Therefore, beamforming designs with only phase variations under constant mode constraints have attracted particular attention.
[0082] Due to the constant modulus constraint, this problem becomes non-convex and NP-hard, making direct solutions difficult to implement. Existing algorithms typically address this problem indirectly through relaxation, as seen in references such as: "X. Zhang, Z. He, X. Zhang and W. Peng, High-performance, beam pattern synthesis via linear fractional semidefinite relaxation, and quasi convex optimization, IEEE Transactions on Antennas, Propag., vol. 66, no. 7, pp. 3421-3431, Jul. 2018." (Reference 1); "Y. Lee, Adaptive Interference Suppression of Phase-Only. Thinned Arrays via Convex Optimization," in IEEE Transactions on Antennas and Propagation, vol. 68, no. 6, pp. 4583-4592, June 2020." (Reference 2); and "J. Liang, X. Fan, HC So and D. Zhou, Array beam pattern synthesis without specifying lobe level masks, IEEE Transactions on Antennas and Propagation, vol. 68, no. 6, pp. 4583-4592, Jul. 2020." (Reference 2); and "J. Liang, X. Fan, HC So and D. Zhou, Array beam pattern synthesis without specifying lobe level masks, IEEE Transactions on Antennas and Propagation, vol. 68, no. 6, pp. 4583-4592, Jul. 2020." (Reference 2). Trans.AntennasPropag., vol.68, no.6, pp.4526-4539, Jun.2020.》(Document 3),《Y.Zhuang, Propagation,vol.69,no.10,pp.7053-7058,Oct.2021.》(Document 4)
[0083] The first type of solution addresses the problem by relaxing the constant modulus constraint. Reference 1 uses the semidefinite relaxation (SDR) technique to relax the constant modulus constraint (CMC) and then solves it using the interior point method. However, due to the need for matrix inverse calculation, a large amount of computation is required in each iteration. To solve this problem, Reference 2 proposes a convex relaxation (CR) method by relaxing the constant modulus constraint into an unequal constraint, but it still does not yield a strictly constant modulus solution.
[0084] The second type of solution addresses the issue by relaxing the cost function. Reference 3 proposes the Alternating Direction Multiplier (ADMM) algorithm by decomposing the original problem into multiple tractable subproblems. However, the convergence speed of this method heavily depends on the choice of the penalty factor. To address this, Reference 4 relaxes the single-phase design problem into a two-phase design problem, deriving the Dual Phase Shifter (DPS) method. However, since the original problem in these algorithms has been relaxed or modified, strict constant modulus constraints are no longer guaranteed.
[0085] Unlike existing methods that solve this problem through relaxation, this application proposes a direct method that does not require relaxation. In the proposed method, the original problem is first transformed into an unconstrained problem on a complex circular manifold. To solve the transformed problem, the gradient descent direction and step size that guarantee no increase in the cost function are derived, and a conjugate gradient (RMOCG) algorithm based on the conjugate Riemannian manifold optimization method is proposed.
[0086] This application provides a conjugate gradient beamforming generation method based on manifold optimization, comprising the following steps:
[0087] Step 1: Set initial values, initial descent direction, and initial wavelength.
[0088] Step 2: Initialize i = 1, calculate
[0089] Step 3: Update the Riemann gradient via (10)
[0090] Step 4: Update the Polak-Ribiere conjugate gradient descent direction η via (14)-(16) i
[0091] Step 5: Update step size τ via (18) i .
[0092] Step 6: Update the feasible solution w via (19)-(20) i+1 , and let i=i+1.
[0093] Step 7: If (where ε is a parameter that controls convergence), the optimal solution of the output waveform.
[0094] Specifically, this application provides an example embodiment, and for ease of description, the following definitions are made first:
[0095] The steering vector of a one-dimensional M-element antenna array is
[0096]
[0097] Where rm m = 1, 2, 3, ..., M is the distance between the m-th array element and the first array element.
[0098] Indicates the angle at which the signal is transmitted or received, where λ is the wavelength;
[0099] Let w = [w(1), w(2), ..., w(M)] T ∈C M×1 Let w(m) be the weighting vector of the antenna array, where m = 1, 2, ..., M represents the weighting factor of the m-th element, and satisfies the constant modulus constraint.
[0100] |w(m)|=1,m=1,2…M (2)
[0101] Therefore, the desired energy signal in the θ direction after filtering can be expressed as:
[0102] G(θ) = |w H a(θ)| 2 =w H a(θ)a(θ) H w = w H x(θ)w (3)
[0103] in,
[0104] After filtering at θ i The signal energy of the nearby interference signal is
[0105]
[0106] in,
[0107] θ i The angle of arrival of the interfering signal.
[0108] Θ i It is θ i The range of error angles centered on;
[0109] To minimize the interference signal energy while ensuring the desired energy as much as possible, the problem model for minimizing the interference signal energy is obtained, that is, the problem can be expressed as follows:
[0110]
[0111] Where A0 is the energy of the desired signal.
[0112] Due to the constant modulus constraint, problem (5) is nonconvex and NP-hard. In general, existing methods solve the problem indirectly by relaxing it into a more tractable form. However, these methods can affect performance or introduce huge computational complexity. Therefore, this application proposes a direct method to solve the problem effectively.
[0113] The proposed method first transforms the objective function; then, a direct manifold optimization-based conjugate gradient (ROCG) algorithm is derived to solve the problem, specifically through the following steps:
[0114] By utilizing the computational properties of the Kronecker product and the Lagrange multiplier method, the original problem is transformed into a problem on a complex circular manifold.
[0115] Geometrically, the CM constraint is interpreted as a solution to a smooth Riemannian complex circular manifold that is constrained within a manifold, and its expression is:
[0116] Ω={w∈C M ||w(m)|=1,m=1,2,…M} (6)
[0117] Transforming the problem into an optimization problem on the manifold Ω, problem (6) can be rewritten as follows:
[0118]
[0119] Using the Lagrange multiplication method, problem (7) can be written as
[0120]
[0121] Where λ is the Lagrange multiplier, λ>0, problem (8) can be further simplified to
[0122]
[0123] in
[0124] The problem in equation (9) is an unconstrained optimization problem on a complex circular manifold, which can be solved by gradient descent. Therefore, this application derives the conjugate gradient (RMOCG) algorithm based on manifold optimization to effectively solve (9) directly. The algorithm is as follows: First, calculate the Riemann gradient; second, choose the descent direction, usually the negative direction of the gradient; third, find a suitable step size based on the descent direction; finally, update the data iteratively until the convergence condition is met.
[0125] Step A: Obtain the Riemann gradient:
[0126] The Riemann gradient is obtained by projecting the Euclidean gradient onto the tangent space, and in the i-th iteration, the Riemann gradient is...
[0127]
[0128] in,(·) T w represents the transpose of an element i Let be an arbitrary given point in space Ω, where w i The tangent vector at point Ω is defined as passing through w on Ω. i A vector that is tangent to any smooth curve. It is a tangent space, formed by the midpoint w of the point space Ω. i Composed of all tangent vectors;
[0129]
[0130] Where R(·) is the real part, and ⊙ represents the Hadamard matrix multiplication;
[0131] The projection operator from Riemannian space to the set of tangent points is expressed as follows:
[0132]
[0133] The Euler gradient is represented as follows:
[0134]
[0135] Step B: Choosing the descent direction η: The Polak-Ribiere conjugate gradient descent direction contains second-order information, which guarantees the superlinear convergence of the algorithm. Therefore, the Polak-Ribiere conjugate gradient descent direction was chosen.
[0136] The descent method for the i-th iteration is as follows:
[0137]
[0138] in, The vector transfer operation required to add or subtract points in different tangent spaces is represented as:
[0139]
[0140] β i PR These are the Polak-Ribieres conjugate parameters, expressed as
[0141]
[0142] Step C: Find the step size
[0143] The step size is found using the Armijo line search strategy, which ensures that the cost does not increase with each iteration. The smallest integer h ≥ 0 is found that satisfies the following condition:
[0144]
[0145] Where σ,β∈(0,1),τ i-1 >0, the step size of the i-th iteration is
[0146] τ i =τ i-1 β h (18)
[0147] Step D: Update the obtained points: First, in tangent space A provisional point is obtained above, denoted as
[0148]
[0149] Then, substituting the found points back onto the complex manifold circle, it can be represented as:
[0150]
[0151] in, is Hadamard element-wise deviation, It is the modulus of this element.
[0152] The computational complexity analysis of the conjugate gradient algorithm based on manifold optimization, including the complexity analysis of each iteration, is as follows:
[0153] Calculating the Riemann gradient requires (M) 2 +2M) multiplication operations; calculating the descent direction requires (5M) multiplication operations and M division operations; calculating the step size requires ((h+1)M 2 +(h+1)M) multiplication operations; updating requires M multiplication operations and M division operations.
[0154] Therefore, the total complexity required for one iteration is ((h+2)M) 2 +(h+11)M), where h<<M.
[0155] In another example implementation's simulation scenario, consider a uniform linear array with M = 64 antennas, a target direction of θ0 = 6°, and an interference direction of θ. i ={-59°,-58°,-57°,-56°,-55°,-34°,-33°,-32°,-31°,-30°}.
[0156] Figure 1The diagram shown is a convergence curve of the objective function of the conjugate gradient beamforming generation method based on manifold optimization provided in an example embodiment of this application. Figure 1 As shown, the convergence performance curve of the objective function in this application is displayed. It can be seen that the convergence performance of this application is good.
[0157] Figure 2 A comparison diagram of the notch depth for a conjugate gradient beamforming generation method based on manifold optimization, provided as another example embodiment of this application. (See diagram for reference.) Figure 2 As shown, the performance of this application is compared with that of existing method 1 (Document 2) and existing method 2 (Document 4). The notch obtained by the present invention is 5 dB and 10 dB lower than that of existing method 1 and existing method 2, respectively. Existing method 1 (Document 2) is represented by DPS in the figure, and existing method 2 (Document 4) is represented by CR in the figure. Therefore, the proposed method is superior to existing method 1 and existing method 2.
[0158] Figure 3 A comparison of runtime for a manifold optimization-based conjugate gradient beamforming generation method, provided as another example embodiment of this application. (See diagram below.) Figure 3 As shown, the runtime of this application is one order of magnitude faster than existing methods 1 and 2. (From top to bottom) Figure 3 The methods shown are, in order: CR method, DPS, and Propose method. Therefore, the technical solution proposed in this application has a relatively fast running speed.
[0159] Those skilled in the art will understand that embodiments of this application can be provided as methods or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.
[0160] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.
[0161] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.
[0162] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.
[0163] In a typical configuration, a computing device includes one or more processors (CPUs), input / output interfaces, a network interface, and memory. Memory may include non-persistent storage in computer-readable media, such as random access memory (RAM) and / or non-volatile memory, such as read-only memory (ROM) or flash RAM. Memory is an example of computer-readable media.
[0164] Computer-readable media includes both permanent and non-permanent, removable and non-removable media that can store information using any method or technology. Information can be computer-readable instructions, data structures, modules of programs, or other data. Examples of computer storage media include, but are not limited to, phase-change memory (PRAM), static random access memory (SRAM), dynamic random access memory (DRAM), other types of random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EEPROM), flash memory or other memory technologies, CD-ROM, digital versatile optical disc (DVD) or other optical storage, magnetic tape, magnetic magnetic disk storage or other magnetic storage devices, or any other non-transferable medium that can be used to store information accessible by a computing device. As defined herein, computer-readable media does not include transient computer-readable media, such as modulated data signals and carrier waves.
[0165] It should also be noted that the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus. Unless otherwise specified, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes that element.
[0166] The above are merely embodiments of this application and are not intended to limit the scope of this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the scope of the claims of this application.
[0167] Other embodiments of this application will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of this application that follow the general principles of this application and include common knowledge or customary techniques in the art not disclosed herein. The specification and examples are to be considered exemplary only, and the true scope and spirit of this application are indicated by the following claims.
[0168] It should be understood that this application is not limited to the precise structure described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope. The scope of this application is limited only by the appended claims.
Claims
1. A method for generating conjugate gradient beamforming based on manifold optimization, characterized in that, include: Construct a problem model that minimizes the energy of the interference signal while ensuring the desired energy; By transforming the objective function, the problem model of minimizing interference signal energy is converted into a problem model on a complex circular manifold. The problem model on the complex circular manifold is solved using the gradient descent method. Initial waveform values are called, and the output waveform with the optimal desired signal energy in the interference direction is solved. The problem model of minimizing interference signal energy is converted into a problem model on a complex circular manifold by utilizing the computational properties of the Kronecker product and the Lagrange multiplier method, including: The constant modulus constraint problem in the problem model of minimizing interference signal energy is transformed into an unconstrained problem model on a complex circular manifold. The conjugate gradient algorithm based on manifold optimization is derived, and the unconstrained problem model on the complex circular manifold is solved using the conjugate gradient algorithm. For the unconstrained optimization problem on a complex circular manifold, the conjugate gradient algorithm based on manifold optimization is used for solution, including: Calculate the Riemann gradient; Choose the descent direction; Find the appropriate step size based on the direction of descent; Finally, the data is continuously updated using an iterative method until the convergence condition is met.
2. The method according to claim 1, characterized in that, The problem model for minimizing the energy of the interference signal while ensuring the desired energy includes: For a multidimensional antenna array, generate an antenna array weighted vector that satisfies the constant modulus constraint; Calculate the expected signal energy in the direction of interference after filtering, and construct a problem model that minimizes the interference signal energy while ensuring the expected energy.
3. The method according to claim 2, characterized in that, The desired signal energy in the filtered interference direction is obtained by substituting the minimum interference signal energy that satisfies the convergence condition of the conjugate gradient algorithm, and then the optimal output waveform in the interference direction is solved by substituting the initial descent direction and the initial wavelength.
4. The method according to claim 3, characterized in that, By transforming the objective function, the problem of minimizing the energy of the interference signal is converted into a problem model on a complex circular manifold. The detailed steps include: The steering vector of a one-dimensional M-element antenna array is (1) in , Let m be the distance between the m-th array element and the first array element. Indicates the angle at which the signal is transmitted or received. Wavelength; Here is the weighted vector of the antenna array, where Let represent the weighting factor of the m-th element, and satisfy the constant modulus constraint. (2) After filtering The desired energy signal in the direction can be expressed as (3) in, ; After filtering The signal energy of the nearby interference signal is (4) in, , The angle of arrival of the interfering signal. Therefore Given the error angle range centered on the target, and minimizing the interference signal energy while ensuring the desired energy as much as possible, the problem model for minimizing the interference signal energy is as follows: (5) in The energy of the desired signal.
5. The method according to claim 4, characterized in that, The process of transforming the objective function to convert the problem model of minimizing the interference signal energy into a problem model on a complex circular manifold includes: Transform the problem model of minimizing interference signal energy into a manifold. Optimization issues: (6) (7) Using the Lagrange multiplication method, problem (7) can be written as (8) in It is a Lagrange multiplier. Problem (8) can be further simplified to (9) in ; The problem in equation (9) is an unconstrained optimization problem on a complex circular manifold.
6. The method according to claim 5, characterized in that, The detailed steps of the conjugate gradient algorithm based on manifold optimization are as follows: Step A: Obtain the Riemann gradient: The Riemann gradient is obtained by projecting the Euclidean gradient onto the tangent space, and in the i-th iteration, the Riemann gradient is... (10) in, Indicates the transpose of an element. It is space Given an arbitrary point, in The tangent vector at point is defined as at... Up through A vector that is tangent to any smooth curve. It is a tangent space, composed of point spaces. midpoint Composed of all tangent vectors; (11) in, It is the real part. Represented as Hadamard matrix multiplication; The projection operator from Riemannian space to the set of tangent points is expressed as follows: (12) The Euler gradient is represented as follows: (13) Step B: Choose the descent direction The Polak-Ribiere conjugate gradient descent direction contains second-order information; The descent method for the i-th iteration is as follows: (14) in, The vector transfer operation required to add or subtract points in different tangent spaces is represented as: (15) These are the Polak-Ribieres conjugate parameters, expressed as (16) Step C: Find the step size Find the step size using the Armijo line search strategy, and then find the smallest integer that satisfies the following conditions. : (17) in, The step size of the i-th iteration is （18） Step D: Update the obtained points: First, in tangent space A provisional point is obtained above, denoted as (19) Then, substituting the found points back onto the complex manifold circle, it can be represented as: （20） in, is Hadamard element-wise deviation, It is the modulus of this element.
7. The method according to claim 6, characterized in that, It also includes a computational complexity analysis of the conjugate gradient algorithm based on manifold optimization, including the complexity analysis for each iteration as follows: Calculating the Riemann gradient requires This is the first multiplication operation; Calculating the descent direction requires Multiplication operations and This is a division operation; calculating the step size requires... This multiplication operation; update required. Multiplication operations and This is the second division operation.