Optimization method for realizing array beam main lobe width maximization based on relaxation strategy

Abstract

The invention discloses an optimization method for realizing array beam main lobe width maximization based on a relaxation strategy. Relates to the field of wide-beam array antennas. The method comprises the following steps: by taking the minimum width of a transition zone area between a main lobe and a side lobe as an optimization target, constructing an optimization model based on array excitation mapping matrixes of a beam main lobe area and a side lobe area, an expected minimum main lobe power gain value, an upper limit of an iteration increment of a to-be-solved target variable xk and an upper limit of a side lobe level; and solving the constructed optimization model by adopting an iterative updating mode, and when an iteration ending condition is met, obtaining array element excitation based on xk obtained by final solving. According to the optimization method provided by the invention, the non-convex problem of maximizing the main lobe width can be converted into a series of convex optimization sub-problems based on the Taylor expansion and relaxation optimization method under the constraint condition of satisfying the expected minimum main lobe power gain and the side lobe level value; and main lobe width expansion and main lobe power gain value adjustment are synchronously realized by adopting an iterative algorithm.

Description

Technical Field

[0001] The present invention relates to the field of wide - beam array antennas, and particularly to an optimization method for maximizing the main lobe width of an array beam based on a relaxation strategy. Background Art

[0002] Wide - beam array antennas have been widely used in the field of satellite communication, such as the Asia - Pacific satellite system, due to their wide - range signal coverage ability. In this application background, even a slight expansion of the main lobe width of the beam can significantly increase the regional coverage area. However, when the desired minimum main lobe power gain and sidelobe level are limited, the current research on how to maximize the main lobe width of the array beam (Array mainlobe beamwidth maximization, AMBM) is relatively limited.

[0003] The current methods for maximizing the main lobe width of the array beam are mainly based on two categories: based on shaped - beam pattern synthesis (SBPS) and based on power - gain pattern synthesis (PGPS). However, when using the SBPS method to expand the main lobe width of the beam, it is often impossible to ensure the desired minimum main lobe power gain and sidelobe level. While the PGPS - based method can obtain a wider main lobe while satisfying the main lobe power gain value and sidelobe level, the process of obtaining the maximum value of the main lobe width is to use the bisection method to find the maximum value of the width when satisfying the above constraints under the condition of maximizing the minimum value of the main lobe power gain. This will, on the one hand, make the element excitations in each step lack correlation, and on the other hand, it does not directly solve the optimization problem with the goal of broadening the main lobe, thus limiting the further broadening of the main lobe and not ensuring that the obtained main lobe width is the optimal value. Summary of the Invention

[0004] In view of the deficiencies of the prior art, the present invention proposes an optimization method for maximizing the main lobe width of an array beam based on a relaxation strategy, which also simultaneously realizes the expansion of the main lobe width and the adjustment of the main lobe power gain value.

[0005] The technical solution adopted by the present invention is as follows:

[0006] An optimization method for maximizing the main lobe width of an array beam based on a relaxation strategy, comprising the following steps:

[0007] Step 1, construct an optimization model:

[0008]

[0009] Wherein, Θ TBIndicates the transition zone area between the main lobe and the sidelobe; C Θ () represents the complement operation, Θ ML 、Θ SL represent the main lobe region and the sidelobe region of the beam respectively; x Δ represents the iteration increment of the variable related to the array element excitation (x w ); x k represents the iterative variable related to the array element excitation (i.e., the initial iterative point); represents the m-th discrete spatial angle θ m of the main lobe region of the array excitation mapping matrix; represents the s-th discrete spatial angle θ s of the sidelobe region of the array excitation mapping matrix; G min represents the desired minimum main lobe power gain value; Δ max represents ||x Δ ||2 upper limit value; ρ represents the upper limit of the set sidelobe level; the superscript "H" represents the conjugate transpose operation, real() represents the real part operation;

[0010] Step 2, solve the optimization model constructed in Step 1 by means of iterative update, based on the set initial value of x k , according to the iteration increment x Δ obtained by each iteration to update x k : x k =x k +x Δ ;

[0011] When the set iteration end condition is satisfied, based on the finally solved x k get the array element excitation w = Q -1 x k , where the non-singular matrix Q is calculated based on the auxiliary matrix B about the array factor: B = Q H Q, and where a(θ) represents the array factor of all array elements, θ represents the spatial angle.

[0012] Furthermore, the optimization model in Step 1 can be replaced by:

[0013]

[0014] where the relaxation vector t = [t1,..., t L T , the superscript T represents the transpose operation, L represents the number of discrete spatial angles θ l contained in the set transition zone area, represents the l-th discrete spatial angle θ l ​Array excitation mapping matrix.

[0015] Further, the optimization model in step 1 can be replaced by:

[0016]

[0017] where the auxiliary vector q = [q1, q2,..., q L T , the first half of its elements is set to q l1 = l1, the second half of its elements are all q l2 = L - l2, Matrix where the matrix is an integer set, and its elements are:

[0018] For the optimization model introducing the auxiliary vectors q and t, step 2 specifically includes:

[0019] Step 201, set the desired minimum main lobe power gain G min , the maximum number of iterations I m , the upper limit value Δ max , the upper limit of the sidelobe level ρ, the reduction ratio η (0 < η < 1), and the main lobe, transition band, and sidelobe regions Θ ML , Θ TB , Θ SL ;

[0020] Set the initial value of the number of iterations n of the first - layer loop to 1, and set the number of iterations I of the third - layer loop to 1; ||x Δ ||2 convergence threshold δ;

[0021] Step 202, define L n as the width of the main lobe |Θ ML | when the first - layer loop is performed for the nth time, and at the same time let L1 = |Θ ML |, L0 = 0;

[0022] Step 203, start the first - layer loop, when L n > L n-1 :

[0023] Update x based on the PGPS method k and obtain the minimum value G0 of the main lobe power gain in the current set region, and let G = G0;

[0024] Step 204, start the second - layer loop, when G ≥ G min :

[0025] According to the current Θ TB ​Set the matrices R, U, and the vector q;

[0026] Step 205, the third - layer loop starts. When I ≤ I m and ||x Δ ||₂ ≥ δ:

[0027] Solve the constructed optimization model to obtain x for the current iteration Δ and t;

[0028] Delete the elements in t that are less than the set lower limit δ t and readjust the dimension of t; then, according to the dimension of t, reset Θ ML , Θ TB , R, U;

[0029] Let Update x k = x k + x Δ ;

[0030] Let the iteration count I be incremented by 1 and then return to Step 205; until the third - layer loop of this round ends; that is, when I > I m it ends and enters Step 206; or when ||x Δ ||₂ < δ, it ends and enters Step 206;

[0031] Step 206, let I = 1, G = ηG. When G < G min appears for the first time in this round of the second - layer loop, let G = G min ; and compress Θ SL , such that Θ TB returns to the initial width size; return to Step 204. When G < G min appears for the second time in this round of the second - layer loop, end this round of the second - layer loop;

[0032] Step 207, let the iteration count n of the first - layer loop be incremented by 1, and then let L n = |Θ ML |, and return to Step 203 until L n ≤ L n-1 ends the first - layer loop;

[0033] Step 208, take the currently obtained x k as the finally solved x k , and calculate the array - element excitation w = Q -1 x k .

[0034] Furthermore, the order of magnitude of the convergence threshold δ and the lower limit δ t is 10 -3 ~10 -4。

[0035] The technical solution provided by the present invention has at least the following beneficial effects:

[0036] The optimization method proposed by the present invention can, under the constraint conditions of meeting the expected minimum main lobe power gain and sidelobe level value, transform the non-convex problem of maximizing the main lobe width into a series of convex optimization sub-problems based on Taylor expansion and relaxation optimization methods, and adopt an iterative algorithm to simultaneously achieve the expansion of the main lobe width and the adjustment of the main lobe power gain value.

[0037] Considering that the existing method for maximizing the main lobe width of a beam based on SBPS is difficult to ensure the expected minimum main lobe power gain and sidelobe level value, the present invention is based on the PGPS method to maximize the main lobe width under the condition of meeting the corresponding constraints;

[0038] Considering that the existing method for maximizing the main lobe width of a beam based on PGPS is difficult to maintain the correlation of element excitations during the algorithm iteration process; and can only indirectly find the maximum value of the main lobe width when meeting the constraint conditions, the present invention takes expanding the main lobe as the direct objective of the optimization solution, adopts an iterative algorithm with a three-layer loop structure, ensures the inheritance and transmission of element excitations in adjacent iteration steps, and simultaneously expands the main lobe width and reduces the main lobe power gain to achieve the maximization of the main lobe width. Description of the Drawings

[0039] The above and / or additional aspects and advantages of the present invention will become apparent and easy to understand from the following description of the embodiments in conjunction with the drawings, where:

[0040] Figure 1 is a schematic flowchart of an optimization method for realizing the maximization of the main lobe width of an array beam based on a relaxation strategy provided by an embodiment of the present invention;

[0041] Figure 2 is the main lobe broadening process when executing loops A and B in 4 different G min cases;

[0042] Figure 3 is the main lobe broadening results when executing loops A and B and loops A, B, and C in the optimization method proposed by the present invention respectively in 4 different G min and initial main lobe width cases;

[0043] Figure 4 is the power gain diagrams obtained by the optimization method proposed by the present invention and the algorithm based on SBPS in 4 different G min cases;

[0044] Among them, the 4 G min are 9.0 dBi, 7.0 dBi, 5.0 dBi, and 3.0 dBi in sequence, andFigures 2 - 4 In the correspondence is carried out through a, b, c, d. For example, 2a, 3a, and 4a respectively correspond to G min = 9.0 dBi case. Specific implementation mode

[0045] In order to enable those skilled in the art to better understand the technical solutions in this specification, the technical solutions in the embodiments of the present invention will be described in detail and completely below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the embodiments described by referring to the accompanying drawings are exemplary and are intended to explain the present invention and should not be construed as a limitation to the present invention.

[0046] In order to facilitate the understanding of the optimization method proposed in the embodiments of the present invention, some main formula symbols are described as follows:

[0047] (1) One-dimensional N-element linear array far-field radiation pattern:

[0048] E(θ) = w H a(θ) (1)

[0049] Wherein, w = [w1, w2,..., w N H represents the element excitation, a(θ) = [a1(θ),..., a N (θ)] H represents the array factor, w i , a i (θ) respectively represent the element excitation and array factor of the i-th element, and θ ∈ [0°, 180°] is the spatial angle sampling.

[0050] (2) Composite beam pattern:

[0051] f w (θ) = c|E(θ)| = c|w H a(θ)| (2)

[0052] Wherein, c is the spatial attenuation factor, and since it is independent of the spatial angle, it can usually be ignored.

[0053] (3) Array power gain:

[0054]

[0055] Wherein, the matrix A(θ) = a(θ)a H (θ), matrix

[0056] ​The current methods for maximizing the main lobe width of array beams mainly fall into two categories: SBPS and PGPS. For the method of maximizing the main lobe width by means of SBPS, it is often difficult to ensure the desired minimum main lobe power gain and sidelobe level value; while for the method of maximizing the main lobe width of the beam based on PGPS, since the bisection method is used to search for the optimal value, it is impossible to maintain the correlation of the element excitations during the algorithm iteration and can only find the maximum value of the main lobe width under the constraint conditions in an indirect way, rather than taking the broadening of the main lobe as the direct optimization goal to adjust the element excitations. Among them, the specific processing procedures of SBPS and PGPS are as follows:

[0057] (1) Method for maximizing the main lobe width of array beam based on SBPS

[0058] (1-1) SBPS 11 :

[0059]

[0060] Among them, f d (θ) is the specified expected value, Θ ML and Θ SL represent the main lobe region and the sidelobe region of the beam respectively. The optimization problem shown in formula (4) is to minimize the ripple size ∈ of the main lobe region while ensuring that the maximum value of the sidelobe level does not exceed the given upper limit ρ.

[0061] (1-2) SBPS 12 :

[0062]

[0063] The optimization goal of the optimization problem shown in formula (5) is to minimize the maximum value of the sidelobe level while ensuring that the ripple size of the main lobe region does not exceed the given upper limit ∈. The remaining SBPS 21 and SBPS 22 can also be defined in the above form:

[0064] (1-3) SBPS 21 :

[0065]

[0066] (1-4) SBPS 22 :

[0067]

[0068] It can be seen that according to the different constraint objects, SBPS can be divided into two categories: SBPS 11 and SBPS 12 The constraints in the problem are designed for the amplitude of the radiation pattern; while SBPS21 With SBPS 22 It is to limit the power of the radiation pattern. By exchanging the optimization objectives ∈ and ρ, the above four optimization problems can be obtained.

[0069] (2) Method for maximizing the main lobe width of the array beam based on PGPS

[0070] As can be seen from formula (3), matrix B is a positive definite Hermitian matrix, so it can be represented by a non-singular matrix Q, that is, B = Q H Q. Define x w = Qw as the variable related to the array element excitation. Substituting it into formula (3), we can get Let the array excitation mapping matrix P(θ) = Q -H A(θ)Q -1 , then it is further transformed into the following power gain synthesis problem (the following remains unchanged):

[0071]

[0072] where, x w = Qw is the variable related to the array element excitation, θ m ∈ Θ ML , m = 1, 2,..., M, and θ s ∈ Θ SL , s = 1, 2,..., S. Among them, M and S respectively represent the discrete spatial angle numbers of the set main lobe region and sidelobe region. The optimization objective shown in formula (8) is to maximize the minimum value G0 of the main lobe power gain.

[0073] The main idea adopted by the existing methods for maximizing the main lobe width is that after presetting the main lobe, sidelobe regions and the desired minimum main lobe power gain, a wide beam is obtained based on the SBPS or PGPS method. If the minimum value of the main lobe power gain of this wide beam is greater than the given lower limit value, the main lobe region is increased; if it is less than the given lower limit, the main lobe region is decreased. Then, the corresponding wide beam and its main lobe power gain are obtained using the new main lobe and sidelobe regions. In each iteration process, the increment or decrement of the main lobe width is gradually halved in the form of the dichotomy. Repeat the above steps until the following conditions are met: 1). The change amount of the main lobe width is less than a certain specified lower limit value; 2). The minimum main lobe power gain is not lower than the desired minimum value, and then the iteration ends.

[0074] However, the disadvantages of this processing method are mainly reflected in: 1) The existing method for maximizing the beam main lobe width based on SBPS is difficult to ensure the desired minimum main lobe power gain and sidelobe level value; 2) Although the existing method for maximizing the beam main lobe width based on PGPS can meet the constraint conditions, the bisection method idea adopted by the algorithm is difficult to maintain the correlation of array element excitations during the iteration process and does not take broadening the main lobe as the direct optimization goal when solving the optimization problem.

[0075] In view of the above deficiencies, the embodiments of the present invention propose an optimization method for maximizing the array beam main lobe width based on a relaxation strategy, which can, under the constraint conditions of meeting the desired minimum main lobe power gain and sidelobe level value, transform the non-convex problem of maximizing the main lobe width into a series of convex optimization sub-problems based on Taylor expansion and relaxation optimization methods, and adopt an iterative algorithm to simultaneously realize the expansion of the main lobe width and the adjustment of the main lobe power gain value.

[0076] In order to meet the desired minimum main lobe power gain and sidelobe level, the present invention also proposes a corresponding main lobe width maximization algorithm based on the PGPS method, that is, in the form of problem (8).

[0077] First, construct an optimization model:

[0078]

[0079] Among them, |Θ ML | represents the angular length of the main lobe region, and G min is the desired minimum main lobe power gain value. Denote the transition band region (between the main lobe and the sidelobe) as Θ TB . When the sidelobe region is fixed, if it is required that the main lobe width |Θ ML | reaches the maximum, it means that the width |Θ TB | of the transition band region needs to reach the minimum. In view of this, the above problem is equivalent to the following form:

[0080]

[0081] Among them, C Θ represents the complement operation, and Θ TB ∪ Θ SL represents the union of the transition band region and the sidelobe region.

[0082] Due to the non-convexity of the first constraint, problem (10) is not a convex programming. Based on this, the embodiments of the present invention introduce Taylor expansion to approximate it. For the sake of convenience of representation, the array excitation mapping matrices P(θ m ), P(θ l ), P(θ s ) of the main lobe, transition band, and sidelobe regions are respectively denoted as and Let the first constraint function of Equation (10) be Using Taylor expansion for it can be expressed as:

[0083]

[0084] where, x Δ = x w - x0. real() represents the operation of taking the real part.

[0085] When ||x Δ ||2 is small enough, can be approximated as 0, and in this way Equation (11) can be expressed as:

[0086]

[0087] Since an iterative algorithm is needed to approximate the final x w , let the initial iteration point be denoted as x k , and use the following optimization problem to continuously update it:

[0088]

[0089] where, Δ max represents the upper limit value of ||x Δ ||2. When ||x Δ ||2 is not greater than this value, can be approximated as 0. After obtaining the corresponding x Δ by solving the optimization problem shown in Equation (13), update x k = x k + x Δ , and use this updated x k as the known variable for the next planning and solution.

[0090] Next is about how to quantify the optimization objective |Θ TB | and transform it into a convex function form:

[0091] In this embodiment, a relaxation vector t = [t1,..., t L T is introduced, where L represents the number of discrete spatial angles θ l contained in the transition band. The smaller L is, the more it reflects the width of the transition band region |Θ TB ​|The smaller the value is. The role of the relaxation vector is to measure the difference between the power gain at each spatial angular position included in the transition band and the minimum power gain of the main lobe. When the number of non-zero elements in t is smaller, it indicates that the width of the transition band is smaller. Therefore, by performing the zero-norm operation on the vector t, the number of its non-zero elements can be obtained, and problem (13) can be equivalently expressed as:

[0092]

[0093] However, because ||t|| o is a non-convex function, this problem is still not a convex programming. Considering that in the left transition band region, the corresponding elements in t, that is, t l , is monotonically decreasing, while in the right side, t l , is monotonically increasing, and for any side, t l is non-negative. Therefore, introduce the matrix whose element form is:

[0094]

[0095] where, represents the set of integers.

[0096] Based on the matrix R, continue to define the matrix U:

[0097]

[0098] To further relax ||t||0, also introduce q = [q1, q2,..., q L T , where the first half of the elements q l = l, For the second half of the elements, q l = L - l, Using the matrices and vectors defined above, the optimization problem shown in formula (14) can be transformed into the following form:

[0099]

[0100] Therefore, this originally non-convex optimization problem shown in formula (9) is transformed into a series of convex sub-problems. By solving the optimization problem shown in formula (17), a wide beam with the narrowest transition band can be obtained, and at the junction of the transition band and the main lobe, that is, near the L / 2 position of the vector t, its element value t l will be quite small. When t l is less than a lower limit value (such as 10 -4 ​) It is regarded as 0 and deleted, which means that in this spatial angular position, the main lobe can replace the original transition band, so as to realize the broadening of the main lobe region of the array beam.

[0101] In one embodiment, as Figure 1 shown, an optimization method for maximizing the main lobe width of an array beam based on a relaxation strategy proposed in an embodiment of the present invention includes the following steps:

[0102] Step 1: Taking the minimum width of the transition band region between the main lobe and the side lobe as the optimization goal, based on the array excitation mapping matrices of the main lobe region and the side lobe region of the beam, the desired minimum main lobe power gain value, the target variable x k to be solved, the iteration increment x Δ of x

[0103] and the side lobe level upper limit, an optimization model is constructed as shown in formulas (13), (14), and (17); k Step 2: The constructed optimization model is solved by an iterative update method. When the set iteration end condition is met, based on the finally solved x -1 x k the array element excitation w = Q

[0104] In one embodiment, based on the optimization problem (17), an embodiment of the present invention proposes a corresponding main lobe width maximization algorithm, which mainly involves three nested loops. For the convenience of description, the innermost loop to the outermost loop are denoted as loop A, loop B, and loop C in sequence. The complete algorithm flow is as follows:

[0105] Step 1: Initialize the desired minimum main lobe power gain G min , the maximum number of iterations I m contained in loop A, the current iteration number I = 1, Δ max , the reduction ratios η, ρ, and the main lobe, transition band, and side lobe regions Θ ML , Θ TB , Θ SL ;

[0106] Step 2: Define L n as the width |Θ ML | of the main lobe when loop C is performed for the nth time, and at the same time let L1 = |Θ ML |, L0 = 0, and the initial value of n is 1;

[0107] Step 3: Start loop C. When L n > L n-1 :

[0108] Step 4: Based on the PGPS method, that is, solving the optimization problem (8) to update x kAnd obtain the minimum value \(G_0\) of the main lobe power gain in the current set area, and let \(G = G_0\);

[0109] Step 5: Start loop B. When \(G\geq G\) min :

[0110] Step 6: Set the matrices \(R\), \(U\) and the vector \(q\) according to the current \(\Theta\) TB ;

[0111] Step 7: Start loop A. When \(i\leq i\) m and \(\|x\) Δ \|_2\geq10\) -4 :

[0112] Step 8: Use the convex optimization tool to solve the optimization problem (17) to obtain \(x\) Δ and \(t\);

[0113] Step 9: Delete the elements in \(t\) that are less than the lower limit value (such as \(t\) l \leq10\) -4 ), and readjust the dimension of \(t\);

[0114] Step 10: Reset \(\Theta\) ML , \(\Theta\) TB , \(R\), \(U\)

[0115] Step 11: Let Update \(x\) k =x\) k +x\) Δ ;

[0116] Step 12: Let \(I = I + 1\) and then return to Step 7;

[0117] Step 13: Let \(I = 1\), \(G=\eta G\), where \(0\lt\eta\lt1\). When \(G\lt G\) appears for the first time in this round of loop B min , let \(G = G\) min ;

[0118] Step 14: Compress \(\Theta\) SL , so that \(\Theta\) TB returns to the initial width size; then return to Step 5. When \(G\lt G\) appears for the second time in this round of loop B min , loop B ends;

[0119] Step 15: Let \(n=n + 1\), \(L\) n =|\Theta\) ML |

[0120] Step 16: Return to Step 3 until \(L\) n \leq L\) n-1 and loop C ends;

[0121] Step 17: Calculate the array element excitation \(w = Q\)-1 x k and use it as the return value.

[0122] In the above steps 13 - 14, since G gradually decreases due to the reduction factor η, considering that it cannot be guaranteed that the reduction process makes G exactly reach G min , so step 14 restricts that "when G < G min appears for the first time in the current loop B, take G = G min ", and then perform the last B loop. When G is less than G min for the second time, exit the B loop of this round.

[0123] The optimization method proposed in the embodiments of the present invention can be directly applied to non - uniform or uniform - spacing array structures and can meet any desired minimum main - lobe power gain. After initializing an array wide - beam with a small main - lobe width and a high main - lobe power gain (G0 > G min ), this optimization method can continuously broaden the main - lobe until G = G min . Finally, it can maximize the main - lobe width under the condition of any given minimum value of the desired main - lobe power gain.

[0124] Embodiment 1:

[0125] This embodiment is used to verify whether the optimization method proposed in the present invention can meet the requirement of maximizing the main - lobe width of the array beam and to demonstrate the functions and meanings of loop A, loop B, and loop C. The set array structure is a non - uniform distribution with 41 array elements, and the distribution of the array elements on the positive semi - axis (normalized with respect to the wavelength) is: 0.3749, 0.6299, 1.5302, 1.8494, 2.3497, 2.8973, 3.2995, 3.8098, 4.6065, 5.0000, 5.3749, 5.6299, 6.5302, 6.8494, 7.3497, 7.8973, 8.2995, 8.8098, 9.6065, 10.000. The spatial angle resolution is set to 0.2°, the central angle is 90°, I m = 5, η = 0.95, Δ max = 0.03, ρ = - 15dB. Here, 4 cases of desired minimum main - lobe gains are discussed, namely 9.0dBi, 7.0dBi, 5.0dBi, 3.0dBi, and the corresponding initial main - lobe widths are 3.0°, 3.0°, 10.0°, 10.0° respectively.

[0126] Figure 2 is the process of separately performing loop A and loop B. After initialization, only the steps in loop A and loop B are used to update x k . For the convenience of demonstration, Figure 2Recorded the synthesis results after each round of the last 5 cycles of Loop B (named in sequence as: 'Round 1', 'Round 2', 'Round 3', 'Round 4', 'Round 5'), which is used to show that the proposed method can continuously broaden the main lobe until G = G min ; Figure 3 Based on Loop A and Loop B, it further calls the steps of Loop C to broaden the main lobe. The curves respectively show the maximum main lobe width results in two cases of whether Loop C is called or not, aiming to verify the importance of Loop C for this optimization method. Since the main considerations are the main lobe power gain and the main lobe width, the display of the side lobes is omitted in the figure. Table 1 further shows the comparison of the maximum main lobe width results under various G min cases.

[0127] Table 1 The maximum main lobe width results obtained with or without executing Loop C under different G min cases

[0128]

[0129]

[0130] Example 2:

[0131] In this example, by comparing the results of the optimization method proposed in the present invention and the existing method in terms of maximizing the main lobe width, the performance advantages of the proposed method are reflected.

[0132] (1) Performance comparison between the optimization method proposed in the present invention and the existing SBPS-based algorithms:

[0133] There are mainly 4 types of SBPS-based algorithms, namely SBPS 11 , SBPS 12 , SBPS 21 , SBPS 22 , and the parameters involved can be set as ξ = -18dB, ∈ = 0.5055dB; for the optimization method proposed in the present invention, let ρ = -12dB, η = 0.955, Δ max = 0.03. The spatial angle resolution is uniformly set to 0.1°, and the array structure is the same as that in Example 1. Figure 4 Table 2 and Table 2 discuss 4 expected minimum main lobe power gain G min cases, namely 10.0dBi, 8.0dBi, 6.0dBi, 4.0dBi, and show the performance differences between the two types of processing methods in achieving the maximum main lobe width. It can be seen that the optimization method proposed in the present invention is significantly superior to the existing SBPS-based algorithms in terms of maximizing the main lobe width.

[0134] Table 2 The maximum main lobe width results under different G minIn this case, the main lobe widths obtained by the proposed method and the existing SBPS-based algorithms

[0135] <![CDATA[G min > Proposed method <![CDATA[SBPS 11 > <![CDATA[SBPS 12 > <![CDATA[SBPS 21 > <![CDATA[SBPS 22 > 10.0 dBi 8.6° 6.0° 6.2° 5.4° 7.2° 8.0 dBi 15.4° 11.2° 8.6° 12.0° 13.8° 6.0 dBi 26.0° 22.8° 10.6° 20.8° 21.0° 4.0 dBi 43.6° 35.2° 14.0° 40.4° 39.2°

[0136] (2) Performance comparison between the optimization method proposed in the present invention and the existing PGPS-based algorithms:

[0137] Both the optimization method proposed in the present invention and the existing PGPS-based algorithms use the optimization problem (8), so some parameters can be shared. Let |Θ TB | = 10°, ρ = -12 dB as the unified parameter values for both. For the other parameter values required for the optimization method proposed in the present invention, they are kept consistent with those in (1). Table 3 discusses 15 different G min cases and records the results of the maximum main lobe widths obtained by the two types of algorithms respectively. It can be seen that the main lobe width obtained by the optimization method proposed in the present invention is always greater than or equal to that of the existing PGPS-based algorithms. Specifically, in the 11 cases where G min = 10.0 dBi, 9.5 dBi, 9.0 dBi, 8.5 dBi, 8.0 dBi, 7.5 dBi, 5.5 dBi, 4.5 dBi, 4.0 dBi, 3.5 dBi, 3.0 dBi, the proposed method has further improved performance compared with the existing method.

[0138] Table 3 Main lobe widths obtained by the proposed method and the existing PGPS-based algorithms in different G min cases

[0139] <![CDATA[G min > 10.0 dBi 9.5 dBi 9.0 dBi 8.5 dBi 8.0 dBi Proposed method 8.6° 10.2° 11.6° 13.4° 15.4° Existing algorithm 8.4° 9.8° 11.4° 13.2° 15.2° <![CDATA[G min > 7.5 dBi 7.0 dBi 6.5 dBi 6.0 dBi 5.5 dBi Proposed method 17.6° 20.0° 22.8° 26.0° 29.6° Existing algorithm 17.4° 20.0° 22.8° 26.0° 29.4° <![CDATA[G min > 5.0 dBi 4.5 dBi 4.0 dBi 3.5 dBi 3.0 dBi Proposed method 33.8° 38.4° 43.8° 49.8° 56.8° Existing algorithm 33.8° 38.2° 43.6° 49.6° 56.6°

[0140] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the application and are not intended to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. An optimization method for maximizing the main lobe width of an array beam based on a relaxation strategy, characterized in that including the following steps: Step 1, construct an optimization model: Among them, Θ TB represents the transition zone region between the main lobe and the side lobe; C Θ () represents the complement operation, Θ ML and Θ SL represent the main lobe region and the side lobe region of the beam respectively; x Δ represents the iterative increment of the variable related to the array element excitation; x k represents the iterative variable related to the array element excitation; represents the array excitation mapping matrix of the m-th discrete spatial angle θ m in the main lobe region; represents the array excitation mapping matrix of the s-th discrete spatial angle θ s in the side lobe region; G min represents the desired minimum main lobe power gain value; Δ max represents the upper limit value of ||x Δ ||2; ρ represents the upper limit of the set side lobe level; the superscript "H" represents the conjugate transpose operation, and real() represents the real part operation; Step 2: Solve the optimization model constructed in Step 1 by means of iterative update. Based on the set initial value of x k , according to the iterative increment x Δ obtained by solving in each iteration, update x k : x k = x k + x Δ ; When the set iteration end condition is satisfied, based on the finally solved x k the array element excitation w = Q -1 x k , where the non-singular matrix Q is calculated based on the auxiliary matrix B regarding the array factor: B = Q H Q, and where a(θ) represents the array factor of all array elements, and θ represents the spatial angle.

2. The method according to claim 1, characterized in that Replace the optimization model in Step 1 with: Among them, the relaxation vector t = [t1,..., t L T , the superscript T represents the transpose operation, and L represents the number of discrete spatial angles θ l included in the set transition band region, represents the l-th discrete spatial angle θ l of the array excitation mapping matrix in the transition band region.​ 3. The method according to claim 1, characterized in that Replace the optimization model in Step 1 with: Among them, the relaxation vector \(t = [t_1,...,t L T , where the superscript \(T\) represents the transpose operation, and \(L\) represents the number of discrete spatial angles \(\theta l included in the set transition band region, represents the \(l\)-th discrete spatial angle \(\theta l of the transition band region, and the array excitation mapping matrix of the auxiliary vector \(q = [q_1,q_2,...,q L T , the first half of whose elements are set to and the second half of whose elements are all matrix where the matrix is an integer set, and its elements are: ​​ 4. The method according to claim 3, wherein Step 2 specifically includes: Step 201, set the desired minimum main lobe power gain G min , the maximum number of iterations I m , the upper limit value Δ max , the upper limit of the sidelobe level ρ, the reduction ratio η, and the main lobe, transition band, and sidelobe regions Θ ML , Θ TB , Θ SL ; Set the initial value of the iteration count n of the first - layer loop to 1, and set the iteration count I of the third - layer loop to 1; ||x Δ ||The convergence threshold δ of 2; Step 202, define L n as the width of the main lobe |Θ ML | when the first-layer loop is performed for the nth time, and at the same time let L1 = |Θ ML |, L0 = 0; Step 203, the first layer of loop starts. When L n > L n-1 : Update x based on the PGPS method k And obtain the minimum value G0 of the main lobe power gain under the current set area, and let G = G0; Step 204, the second-layer loop starts. When G ≥ G min : According to the current Θ TB Set the matrices R, U and the vector q; Step 205, the third-layer loop starts. When I ≤ I m and ||x Δ ||2 ≥ δ: Solve the constructed optimization model to obtain x for the current iteration Δ and t; Delete the elements in t that are less than the set lower limit δ t and readjust the dimension of t; then reset Θ ML according to the dimension of t TB , Θ , R, U; Let Update x k = x k + x Δ ; Increment the iteration count I by 1 and then return to step 205; until the third-level loop of this round ends; that is, when I > I m ends and enters step 206; or when ||x Δ ||2 < δ, ends and enters step 206; Step 206, let I = 1, G = ηG, when G appears for the first time in the second cycle of this round <G min When G=G min ; and compression Θ SL , so that Θ TB Restore to the initial width; return to step 204, when G appears for the second time in the second layer loop of this round <G min When , the second cycle of this round ends; Step 207: After incrementing the iteration count n of the first-layer loop by 1, then set L n = |Θ ML |, and return to Step 203 until L n ≤ L n-1 , at which point the first-layer loop ends; Step 208: Take the currently obtained x k as the finally solved x k , and calculate the array element excitation w = Q -1 x k .

5. The method according to claim 4, wherein Convergence threshold δ and lower limit δ t are of the order of magnitude of 10 -3 ~10 -4 .

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