Comprehensive excitation dynamic range optimization method for sparse array forming directional diagram
By establishing a sparse array shaping pattern synthesis model through the alternating direction multiplier method, the sparsity and dynamic range of the excitation are optimized, solving the problem that cannot be optimized simultaneously in the existing technology and improving energy utilization efficiency.
Patent Information
- Application Number
- CN202511394915.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-09-28
- Publication Date
- 2026-01-09
AI Technical Summary
Existing technologies cannot simultaneously and automatically optimize the excitation sparsity and dynamic range of sparse arrays, resulting in low energy utilization efficiency.
An excitation dynamic range optimization model for sparse array shaped pattern synthesis is established using the alternating direction multiplier method. The sparsity and dynamic range of the excitation are optimized by taking the total transmit power as the objective function and combining 0/1-like constraints and shape templates. The solution is obtained using the CVX toolbox.
This approach achieves the optimization of excitation sparsity and dynamic range while meeting the radiation pattern requirements, thereby improving energy utilization efficiency.
Smart Images
Figure CN121301907A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of array signal processing, specifically relating to an excitation dynamic range optimization method for sparse array shaped pattern synthesis. Background Technology
[0002] In pattern synthesis techniques for array antennas, sparse excitation simplifies the array structure and reduces inter-element coupling. Furthermore, minimizing the dynamic range of the excitation (the ratio of the maximum to minimum excitation amplitude for each element) improves energy efficiency. This technique is known as excitation dynamic range optimization in sparse array pattern synthesis. While there is considerable research on sparse array pattern synthesis or excitation dynamic range optimization alone, research on pattern synthesis with both sparse excitation and limited dynamic range is almost nonexistent. Summary of the Invention
[0003] The purpose of this invention is to provide an excitation dynamic range optimization method for sparse array shaped pattern synthesis, which solves the problem that existing methods cannot simultaneously and automatically optimize the sparsity and dynamic range of the excitation.
[0004] To achieve the above objectives, the technical solution adopted by the present invention is as follows:
[0005] The method for optimizing the excitation dynamic range of sparse array-shaped pattern synthesis includes the following steps:
[0006] Step 1: Establish a model for the excitation dynamic range optimization method of sparse array shaped pattern synthesis:
[0007] (1a);
[0008] (1b);
[0009] (1c);
[0010] (1d);
[0011] In the formula, Indicates the dynamic range of the incentive. This represents the minimum non-zero value of the excitation amplitude of the array element. Let be the complex excitation vector of the array. It is the l0 norm. This is the main lobe steering vector of the array. and These are the lower and upper boundaries of the main lobe, respectively. Indicates the main lobe region. For the sidelobe steering vector of the array, This is the upper limit of the preset sidelobe level. Indicates the sidelobe region;
[0012] Step 2: Relax the l0 norm in equation (1a) to the l1 norm, and introduce the following auxiliary variable.
[0013] (2a);
[0014] (2b);
[0015] (2c);
[0016] (2d);
[0017] (2e);
[0018] (2f);
[0019] (2g);
[0020] In the formula, It is the l1 norm. , , and To introduce auxiliary variables, the augmented Lagrangian function is further constructed as follows:
[0021] (3);
[0022] in, In equation (2b) The vector formed In equation (2d) The vector formed and These are the main lobe steering vectors. and side lobe steering vector The guiding matrix formed, , and As dual variables, , and This is a custom iteration step size;
[0023] Step 3: Solve the above problem using the alternating direction multiplier method:
[0024] Step 3.1, Update :
[0025] Ignore and Irrelevant terms can be obtained
[0026] ;
[0027] (4);
[0028] The above equation is transformed into the following subproblems
[0029] ;
[0030] (5);
[0031] in , dual variables The m-th element can be obtained
[0032] (6);
[0033] Step 3.2, Update :
[0034] Ignore and For irrelevant terms, we can obtain:
[0035] ;
[0036] (7);
[0037] The above equation can be transformed into the following subproblems:
[0038] ;
[0039] (8);
[0040] in , dual variables The s-th element can be obtained
[0041] (9);
[0042] Step 3.3, Update
[0043] Ignoring irrelevant terms, we obtain the following optimization problem.
[0044] ;
[0045] (10);
[0046] The above formula is equivalent to
[0047] ;
[0048] (11);
[0049] in, First, solve the following subproblems:
[0050] ;
[0051] (12);
[0052] The above equation can be transformed into the following subproblems:
[0053] ;
[0054] (13);
[0055] in For vectors Take the nth component , can be obtained
[0056] (14);
[0057] The solution to the above equation is
[0058] (15);
[0059] Substituting equation (15) into equation (14), we obtain only those containing quadratic function
[0060] (16);
[0061] in
[0062] (17);
[0063] (18);
[0064] (19);
[0065] Where P1 is in accordance with The set of indices n, P2 is the set of indices n. The set of indices n, P3 is the set of indices n. The set of indices n; and Let P1 and P2 represent the number of elements in set P1 and set P2 respectively; then we have
[0066] (20);
[0067] Will Substituting into equation (15) yields the final result. ;
[0068] Step 3.3, Update :
[0069] Ignore and The terms can be obtained
[0070] (twenty one) ;
[0071] in
[0072] (twenty two) ;
[0073] (twenty three) ;
[0074] The identity matrix is of order N, updated via the CVX toolbox. ;
[0075] Step 3.4, update the dual variable:
[0076] (twenty four) ;
[0077] (25);
[0078] (26);
[0079] Step 4: Iterate through Step 3 until the results of two consecutive iterations are in the same order. If the difference is less than a preset threshold, save the current iteration result. The value of is used to complete the excitation dynamic range optimization of sparse array shaping pattern synthesis.
[0080] Due to the adoption of the above technical solution, the present invention has the following beneficial effects compared with the prior art:
[0081] The model in this invention uses the total transmit power as the objective function to simultaneously optimize the sparsity of the excitation and the non-zero minimum amplitude. It uses 0 / 1-like constraints to limit the sparsity and dynamic range of the excitation, thus solving the problem that existing methods cannot simultaneously and automatically optimize the sparsity and dynamic range of the excitation. Attached Figure Description
[0082] Figure 1 To generate the pattern when the dynamic range D=2;
[0083] Figure 2This refers to the incentive amplitude corresponding to an incentive dynamic range D=2;
[0084] Figure 3 To generate the radiation pattern when the excitation dynamic range D=1.5;
[0085] Figure 4 This refers to the incentive amplitude corresponding to an incentive dynamic range D=1.5;
[0086] Figure 5 The resulting radiation pattern is generated when the excitation dynamic range D = 1.25.
[0087] Figure 6 This represents the incentive amplitude corresponding to an incentive dynamic range D = 1.25.
[0088] Figure 7 To generate the pattern when the dynamic range D=1;
[0089] Figure 8 This is the incentive amplitude corresponding to the dynamic range D=1. Detailed Implementation
[0090] The present invention will now be further described in conjunction with the accompanying drawings and specific embodiments.
[0091] The present invention relates to an excitation dynamic range optimization method for sparse array-shaped pattern synthesis. The technical solution employed is as follows: First, a model for the excitation dynamic range optimization algorithm of sparse array-shaped pattern synthesis is established. This model uses the total transmit power as the objective function to simultaneously optimize the sparsity of the excitation and the non-zero minimum amplitude. Zero / 1-like constraints are used to limit the sparsity and dynamic range of the excitation, and a shape template is used to limit the pattern response. Since this model is highly non-convex and difficult to solve directly, the Alternating Direction Multipliers Method (ADMM) is used to solve the model, obtaining a sparse excitation vector with a limited dynamic range. Specifically:
[0092] Step 1: Establish a model for the excitation dynamic range optimization algorithm of sparse array shaped pattern synthesis.
[0093] Suppose there is a uniform linear array consisting of N elements, whose pattern array factor is...
[0094] (1)
[0095] in, Let be the complex excitation vector of the array, containing the amplitude and phase of the excitation of each array element. The guide vector of the array, The resulting radiation pattern is shown below. A sparse array can be achieved by selecting elements from a uniform linear array, which is equivalent to sparsening the excitation. This can be achieved by optimizing the l0 norm of the excitation. The radiation pattern synthesis of the array can be achieved by constraining the main lobe and side lobes. Therefore, the following model can be constructed:
[0096] (2a)
[0097] (2b)
[0098] (2c)
[0099] (2d)
[0100] in, Indicates the dynamic range of the incentive. This represents the minimum non-zero value of the excitation amplitude of the array element. Let be the complex excitation vector of the array. It is the l0 norm. This is the main lobe steering vector of the array. and These are the lower and upper boundaries of the main lobe, respectively. Indicates the main lobe region. For the sidelobe steering vector of the array, This is the upper limit of the preset sidelobe level. Indicates the sidelobe region. and These represent the angles of the main lobe region and the side lobe region, respectively; Equation (2a) means that the sparsity and amplitude of the excitation are optimized simultaneously with the total transmit power as the objective function, and Equation (2d) is a class of 0 / 1 constraints. Let n represent the excitation value of the nth element in the array. Equations (2a) and (2d) together limit the sparsity and dynamic range of the excitation. Equation (2b) means to make the main lobe approximate the desired radiation pattern. Equation (2c) means to make the side lobes lower than the preset value. Therefore, Equation (2) can optimize the sparsity and amplitude of the excitation while satisfying the directional shape constraints, thereby optimizing the dynamic range of the sparse array shaped pattern comprehensive excitation.
[0101] Step 2, Equivalent transformation of the problem
[0102] The l0 norm in equation (2a) is non-convex and difficult to optimize directly. It can be relaxed to the l1 norm. Constraints (2b) and (2d) are also non-convex. Therefore, equation (2) is a highly non-convex problem, which is difficult to solve directly by optimization. It can be solved using the alternating direction multiplier method. The l0 norm is relaxed to the l1 norm, and the following auxiliary variables are introduced.
[0103] (3a)
[0104] (3b)
[0105] (3c)
[0106] (3d)
[0107] (3e)
[0108] (3f)
[0109] (3g)
[0110] In the formula, It is the l1 norm. , , and This is an auxiliary variable introduced. Further, the augmented Lagrangian function can be constructed as follows.
[0111] (4)
[0112] in, , These are the main lobe steering vectors. and main lobe steering vector The guiding matrix formed, , , As dual variables, , , This is a custom iteration step size.
[0113] Step 3: Solve the above problem using the alternating direction multiplier method.
[0114] (1) Update
[0115] In equation (3b) The vector formed. Ignore the relationship with... Irrelevant terms can be obtained
[0116] (5)
[0117] The above formula can be transformed into M subproblems.
[0118] (6)
[0119] in , can be obtained
[0120] (7)
[0121] (2) Update
[0122] In equation (3b) The vector formed. Similar to step (1), ignoring the vectors... Irrelevant terms can be obtained
[0123]
[0124] (8)
[0125] The above equation can be transformed into S subproblems.
[0126] (9)
[0127] in . We can obtain
[0128] (10)
[0129] (3) Update
[0130] Ignoring irrelevant terms, we obtain the following optimization problem.
[0131] (11)
[0132] The above formula is equivalent to
[0133] (12)
[0134] in, Due to variables and They are coupled together; we can calculate them first. The optimal solution transforms the objective function into one that contains only... The function. First, solve the following subproblems:
[0135] (13)
[0136] The above formula can be transformed into N subproblems.
[0137] (14)
[0138] Pick , can be obtained
[0139] (15)
[0140] The solution to the above equation is
[0141] (16)
[0142] Substituting equation (16) into equation (15), we obtain only those containing quadratic function
[0143] (17)
[0144] in
[0145] (18)
[0146] (19)
[0147] (20)
[0148] Where P1 is in accordance with The set of indices n, P2 is the set of indices n. The set of indices n, P3 is the set of indices n. The set of indices n; and Let P1 and P2 represent the number of elements in set P1 and set P2 respectively; then we have
[0149] (twenty one)
[0150] Further, we can obtain However, it should be noted that equation (16) is used in solving... At that time, the unknown was used. Therefore, alternating optimization is required in this step to obtain the optimal result. .
[0151] (4) Update
[0152] Ignore and The terms can be obtained
[0153] (twenty two)
[0154] in
[0155] (twenty three)
[0156] (twenty four)
[0157] Clearly, the above equation is a convex problem, which can be solved using CVX.
[0158] (5) Update dual variables
[0159] (twenty four)
[0160] (25)
[0161] (26)
[0162] t represents the iteration number. The update process is executed iteratively until the results of two consecutive iterations are in agreement. If the difference is less than a preset threshold, save the current iteration result. The value of is used to complete the excitation dynamic range optimization of sparse array shaping pattern synthesis.
[0163] The following is a more specific example:
[0164] The experiment of this invention uses an 80-element uniform linear array, with the element spacing being 1 / 4 wavelength corresponding to the operating frequency; the experiment employs a flat-top radiation pattern designed in this invention, with the beam center at 0° and the beamwidth at 30°, wherein the main lobe region is... The upper boundary of the main lobe level is set to 0.3 dB, and the lower boundary is set to -0.3 dB. The side lobe region is... The upper bound of the sidelobe level is -14dB. The optimal excitation vector can be obtained through equations (5)-(26). The radiation pattern calculated in this way can achieve the desired radiation pattern while obtaining excitation that is both sparse and has a limited dynamic range. Figure 1 and Figure 2 As shown, the method of this invention can optimize and obtain sparse and dynamically limited excitations when the dynamic range is 2, and only 55 array elements are selected, with a sparsity of 68.75%; similarly, as Figures 3-8 As shown, the method of the present invention can optimize and obtain sparse and dynamically limited excitations when the dynamic range is 1.5, 1.25 and 1 respectively, and only 48, 47 and 43 array elements are selected respectively, with sparsity of 60%, 58.75% and 53.75% respectively.
[0165] This invention optimizes the excitation dynamic range of sparse array shaping pattern synthesis by using the alternating direction multiplier method, thus solving the problem that existing methods cannot simultaneously and automatically optimize the sparsity and dynamic range of the excitation.
Claims
1. A method for optimizing the excitation dynamic range of sparse array shaped pattern synthesis, characterized in that, Includes the following steps: Step 1: Establish a model for the excitation dynamic range optimization method of sparse array shaped pattern synthesis: (1a)； (1b) ； (1c) ； (1d) ； In the formula, Indicates the dynamic range of the incentive. This represents the minimum non-zero value of the excitation amplitude of the array element. Let be the complex excitation vector of the array. It is the l0 norm. This is the main lobe steering vector of the array. and These are the lower and upper boundaries of the main lobe, respectively. Indicates the main lobe region. For the sidelobe steering vector of the array, This is the upper limit of the preset sidelobe level. Indicates the sidelobe region; Step 2: Relax the l0 norm in equation (1a) to the l1 norm, and introduce the following auxiliary variable. (2a) ； (2b) ； (2c) ； (2d) ； (2e) ； (2f) ； (2g) ； In the formula, It is the l1 norm. , , and To introduce auxiliary variables, the augmented Lagrangian function is further constructed as follows: (3) ； in, In equation (2b) The vector formed In equation (2d) The vector formed and These are the main lobe steering vectors. and side lobe steering vector The guiding matrix formed, , and As dual variables, , and This is a custom iteration step size; Step 3: Solve the above problem using the alternating direction multiplier method: Step 3.1, Update : Ignore and Irrelevant terms can be obtained ； (4) ； The above equation is transformed into the following subproblems ； (5) ； in , dual variables The m-th element can be obtained (6) ； Step 3.2, Update : Ignore and For irrelevant terms, we can obtain: ； (7) ； The above equation can be transformed into the following subproblems: ； (8) ； in , dual variables The s-th element can be obtained (9) ； Step 3.3, Update Ignoring irrelevant terms, we obtain the following optimization problem. ； (10) ； The above formula is equivalent to ； (11) ； in, First, solve the following subproblems: ； (12) ； The above equation can be transformed into the following subproblems: ； (13) ； in For vectors Take the nth component , can be obtained (14) ； The solution to the above equation is (15) ； Substituting equation (15) into equation (14), we obtain only those containing quadratic function (16) ； in (17) ； (18) ； (19) ； Where P1 is in accordance with The set of indices n, P2 is the set of indices n. The set of indices n, P3 is the set of indices n. The set of indices n; and Let P1 and P2 represent the number of elements in set P1 and set P2 respectively; then we have (20) ； Will Substituting into equation (15) yields the final result. ; Step 3.3, Update : Ignore and The terms can be obtained (21) ； in (22) ； (23) ； For an N-order identity matrix, update via the CVX toolbox. ; Step 3.4, update the dual variable: (24) ； (25) ； (26) ； Step 4: Iterate through Step 3 until the results of two consecutive iterations are in the same order. If the difference is less than a preset threshold, save the current iteration result. The value of is used to complete the excitation dynamic range optimization of sparse array shaping pattern synthesis.