Conformal Array Transmission Beamforming Method Based on LCMV Complementary Decomposition

Abstract

The present invention relates to radar communication technology, and discloses a conformal array transmit beamforming method based on LCMV complementary decomposition. By constructing an LCMV beamformer, the main beam direction and polarization parameters of the transmit pattern of the polarization conformal array are constrained. Then, the optimal weight vector is subjected to complementary decomposition, and the transmit beam pattern is optimized by adjusting real numbers. It can effectively reduce the peak sidelobe level and cross-polarization level, and form a notch within a set angle range. The present invention can achieve polarization constraint on the transmit beam pattern of the conformal array and effectively perform beamforming on the basis of ensuring that the main beam direction does not shift, so that the transmit beam pattern of the conformal array meets the performance requirements.

Description

Technical Field

[0001] The present invention belongs to the field of radar communication technology, and particularly relates to a conformal array transmit beamforming technology based on LCMV complementary decomposition. Background Art

[0002] A conformal array can be flexibly attached to the surface of a carrier and has the characteristic of conforming to the carrier. Compared with a traditional planar array, it has unique positive characteristics such as easy installation and a larger angular coverage range, and is widely used in fields such as radar, satellites, and communications. However, affected by the curvature of the carrier, it is difficult for a conformal array to ensure that each element has a consistent element pattern and polarization characteristics, which results in a relatively high peak sidelobe level and serious cross polarization in the transmit pattern of the conformal array, seriously affecting the performance of the transmit beam. Therefore, how to determine the weight vector of the conformal array to achieve the desired transmit beam pattern, that is, beamforming, is an important issue.

[0003] With the increasing attention paid to conformal arrays, the theories and algorithms related to beamforming for conformal arrays have been continuously developed. Intelligent optimization algorithms perform beamforming by obtaining the optimal solution through global search using random methods. For example, Li et al. proposed a hybrid optimization algorithm combining genetic algorithm and particle swarm algorithm for beamforming of conformal arrays, which can effectively reduce the peak sidelobe level (see the literature: "A hybrid optimization algorithm and its application for conformal array pattern synthesis", W. Li, X. Shi, Y. Hei, et al., in IEEE Trans. Antennas Propag., 2010, 58, (10), pp. 3401–3406). However, intelligent optimization algorithms have problems such as long time consumption and difficulty in meeting specific performance requirements, and their applications have certain limitations. The alternating projection algorithm is also widely used in conformal array beamforming. By using pattern constraints and the least squares method, it realizes the alternating projection between the desired pattern and the actual pattern, thereby performing beamforming. However, this method is only applicable to relatively simple arrays, and the main beam pointing will shift in complex arrays (see the literature: H. Steyskal, "Pattern synthesis for a conformal wing array," Proceedings, IEEE Aerospace Conference, Big Sky, MT, USA, 2002, pp. 2-2). When the conformal array beamforming problem can be formulated as a convex optimization problem, the convex optimization problem can be solved using a toolkit to obtain the weight vector for pattern design. This method can effectively perform conformal array transmit beamforming, but it has a large amount of calculation, and some algorithms ignore the suppression of cross polarization (see the literature: B. Fuchs and J. J. Fuchs, "Optimal Polarization Synthesis of Arbitrary Arrays With Focused Power Pattern," in IEEE Transactions on Antennas and Propagation, vol. 59, no. 12, pp. 4512-4519, Dec. 2011).

[0004] Therefore, how to synthesize the desired transmit beam pattern of a conformal array by adjusting the response in a specified direction while satisfying the main beam pointing and polarization constraints has important research significance. Summary of the Invention

[0005] The applicant analyzed the advantages and disadvantages of existing classical conformal array transmitting beamforming algorithms. The existing methods fail to meet various performance requirements of the conformal array transmitting beam pattern on the basis of constraining the main beam direction and polarization mode. The technical problem to be solved by the present invention is to propose an algorithm that can ensure the main beam direction and polarization constraints, effectively reduce the peak sidelobe level and cross-polarization level, and form a notch within a set angle range, and can effectively synthesize the conformal array transmitting beam pattern.

[0006] The technical solution adopted by the present invention to solve the above technical problem is a conformal array transmitting beamforming method based on LCMV complementary decomposition. The total number of array elements of the conformal array in the rectangular coordinate system is N. The array elements adopt rectangular microstrip antennas and are placed at horizontal and vertical intervals. The nth array element has an element pattern in the azimuth and elevation directions as where θ is the azimuth variable, is the elevation variable; the spatial steering vector of the conformal array is In the and directions, the steering vectors are respectively and

[0007] The specific steps of the transmitting beamforming are as follows:

[0008] Step 1) Construct a beamformer with linearly constrained minimum variance (LCMV), and set the weight vector w T = R -1 C (C H R -1 C) - 1 f, where C is the constraint matrix, f is the constraint response vector, R is the spatial correlation matrix, · H is the conjugate transpose;

[0009] Step 2) Complementary decompose the weight vector into two components w1 and w2, which are the results weighted by adjusting the parameter β:

[0010] w T = R -1 C (C H R -1 C) -1 f = w1 + βw2;

[0011] The components w1 and w2 are:

[0012]

[0013]

[0014] Among them, To adjust the beam direction, w (0 is the LCMV weight vector w T The initial iteration value of w (0) = C(C H C) -1 f, P ⊥ And P are projection matrices;

[0015]

[0016]

[0017] Among them, represents the steering vector of the conformal array emission pattern adjustment point. By adjusting the components, can be set as the main polarization pattern or the cross-polarization pattern. The middle matrix and the middle parameter α are respectively:

[0018]

[0019]

[0020]

[0021] Among them, I is the identity matrix, p 2 represents the power of the emission pattern adjustment point;

[0022] Step 3) Perform the k-th iteration: Determine the k-th adjustment direction Use the weight vector w obtained from the previous iteration (k-1) The component w in 1,k-1 and w 2,k-1 and the desired level value ρ k Calculate two candidate results β of the adjustment parameter β a and β b :

[0023]

[0024]

[0025] Among them, is to take the real part. B(1,2) represents taking the element in the 1st row and 2nd column of matrix B, and B(2,2) represents taking the element in the 2nd row and 2nd column of matrix B;

[0026]

[0027] The intermediate quantity

[0028] Step 4) Set βa and β b are respectively input into the function F(β) as the adjustment parameter β for calculation, and the one that makes the value of F(β) smaller is selected as the optimal solution β of the adjustment parameter β obtained in the k-th iteration k* , thereby determining the weight vector w obtained in the k-th iteration (k) , w (k) = w 1,k-1 + β k* w 2,k-1 ;

[0029]

[0030] where, ||·||2 is the second norm;

[0031] Step 5) Determine whether the iteration stop condition is reached. If so, use the weight vector w obtained in the k-th iteration (k) as the optimal transmit weight vector w T* for the transmit beamforming of the conformal array; otherwise, update k = k + 1 and return to Step 3); the iteration stop condition is that the transmit pattern determined by the weight vector w obtained in the k-th iteration (k) meets the constraints or k reaches the maximum value.

[0032] The present invention discloses an algorithm for quickly and effectively beamforming the transmit pattern of a conformal array. The present invention cleverly designs an LCMV beamformer to effectively constrain the main beam pointing and polarization information of the transmit pattern of the conformal array. At the same time, an analytical solution of the transmit weight vector is given, the transmit weight vector is complementarily decomposed, and the desired component pattern is effectively adjusted and optimized by adjusting real parameters. And the main polarization and cross-polarization steering vectors are constructed, so that the algorithm can directly adjust the patterns of the main polarization and cross-polarization components under any polarization condition.

[0033] The beneficial effect of the present invention is that while the transmit pattern of the conformal array is aligned with the desired transmit direction and meets the polarization constraints, it can effectively reduce the peak sidelobe level and cross-polarization level of the transmit beam pattern, and form a depression in the specified angle region. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 is a schematic diagram of the angle and coordinate system of the present invention;

[0035] Figure 2 is a flowchart of the method of the present invention;

[0036] Figure 3 is the total initial transmit beam pattern under linear polarization;

[0037] Figure 4 is the initial transmit beam under linear polarization Direction pattern;

[0038] Figure 5 Is the initial emission beam under linear polarization Direction pattern:

[0039] Figure 6 Is the total direction pattern of the emission beams of three algorithms under linear polarization in the present invention;

[0040] Figure 7 Is the top view of the total direction pattern of the emission beams of three algorithms under linear polarization in the present invention;

[0041] Figure 8 Is for three algorithms under linear polarization in the present invention Direction pattern;

[0042] Figure 9 Is for three algorithms under linear polarization in the present invention Direction pattern;

[0043] Figure 10 Is the co-polarization direction pattern of three algorithms under linear polarization in the present invention;

[0044] Figure 11 Is the cross-polarization direction pattern of three algorithms under linear polarization in the present invention;

[0045] Figure 12 Is the elevation plane profile direction pattern of three algorithms under linear polarization in the present invention;

[0046] Figure 13 Is the azimuth plane profile direction pattern of three algorithms under linear polarization in the present invention;

[0047] Figure 14 Is the total direction pattern of the emission beams of three algorithms under circular polarization in the present invention;

[0048] Figure 15 Is the top view of the total direction pattern of the emission beams of three algorithms under circular polarization in the present invention;

[0049] Figure 16 Is for three algorithms under circular polarization in the present invention Direction pattern;

[0050] Figure 17 Is for three algorithms under circular polarization in the present invention Direction pattern;

[0051] Figure 18 Is the co-polarization direction pattern of three algorithms under circular polarization in the present invention;

[0052] Figure 19 Is the cross-polarization direction pattern of three algorithms under circular polarization in the present invention;

[0053] Figure 20 These are the elevation plane radiation patterns of three algorithms under circular polarization in the present invention;

[0054] Figure 21 These are the azimuth plane radiation patterns of three algorithms under circular polarization in the present invention; Detailed implementation manners

[0055] The following further elaborates in detail on the detailed implementation manners and working principles of the present invention with reference to the accompanying drawings.

[0056] For better description, the following definitions are first made:

[0057] Rectangular coordinate system: The positive direction of the z-axis is vertically downward, the y-axis points to the left, and the x-axis is perpendicular to the YOZ plane and points outward.

[0058] Elevation angle and azimuth angle: The elevation angle θ is the angle between the incident signal and the positive semi-axis of the z-axis, and the azimuth angle is the angle between the projection of the incident signal on the XOY plane and the positive semi-axis of the x-axis. The angles and the coordinate system schematic diagram are as Figure 1 shown. The range of the elevation angle is θ ∈ [0, π / 2], and the range of the azimuth angle is

[0059] Hemispherical conformal array: The vertex of the hemispherical surface is located on the positive semi-axis of the z-axis, the center of the sphere coincides with the origin, the radius of the pseudo-hemispherical surface is R, and the array elements are arranged along concentric rings on the hemispherical surface, and are alternately placed parallel and perpendicular to the tangent of the ring.

[0060] Conformal array transmitting radiation pattern: In the far field at and directions, a dipole is placed at each direction, then the spatial responses on the two dipoles are respectively and E θ , and the transmitting beam radiation pattern of the array can be expressed as

[0061]

[0062] Its amplitude is E all is the total transmitting beam radiation pattern.

[0063] Cross-polarization level: The cross-polarization level expression is which describes the cross-polarization component level value normalized based on the main polarization radiation pattern, where E x is the cross-polarization component radiation pattern, and E co is the main polarization component radiation pattern. The smaller the CPL value, the smaller the cross-polarization component relative to the main polarization component. At this time, the antenna polarization is closer to the desired polarization, and thus the antenna shows better working performance in the system.

[0064] Normalized radiation pattern: The expression for the normalized radiation pattern of a conformal array is and The normalized radiation patterns in the

[0065] Rectangular microstrip antenna: The array element uses a rectangular microstrip patch antenna. The microstrip patch antenna is located in the YOZ plane. The length of the rectangle is parallel to the y-axis. The radiation pattern is as described by the following formula

[0066]

[0067] E θ = 0

[0068] Where and E θ are respectively and The radiation patterns of the antenna in the

[0069] Element unit pattern: According to the position of the element, the direction of the nth element in the local coordinate system can be obtained by using Euler rotation Considering the shielding effect of the element carrier and the radiation pattern of the rectangular microstrip patch antenna, the element pattern of the element in the local coordinate system can be obtained:

[0070]

[0071]

[0072] is the and polarization patterns in the local coordinate system of the nth element.

[0073] Finally, the element pattern g in the global coordinate system can be obtained through Euler rotation n,θ ,

[0074] The following specifically describes the embodiments of the present invention in detail with reference to the accompanying drawings of the specification. Assume that the number of array elements is N, the emission beam direction of the conformal array is The polarization parameters are (γ, η), where γ is the polarization angle and η is the phase difference; the array spatial domain steering vector is In and directions, the steering vectors are respectively and

[0075] Such as Figure 2Flowchart of the conformal array transmit beamforming algorithm based on LCMV complementary decomposition, which specifically includes the following steps:

[0076] Step 1: According to the transmit direction and the positions of the conformal array elements, define the array spatial domain steering vector

[0077]

[0078] where λ is the signal wavelength, (x n , y n , z n ) is the position of the nth element. We can obtain and The steering vectors in the and directions are respectively is and The element pattern matrix in the

[0079]

[0080]

[0081] Then and The spatial domain responses in the directions are respectively

[0082] where w is the transmit weight vector. co , η co ) and the cross-polarization parameters (γ x , η x ). Usually, (γ co , η co ) = (γ, η). Define the co-polarization steering vector and the cross-polarization steering vector

[0083]

[0084]

[0085] The spatial domain responses in the co-polarization direction and the cross-polarization direction are respectively It can be found that the conformal array transmit beam pattern can be expressed as

[0086]

[0087] Step 3, Determine the transmission signal direction To align the main lobe with the desired transmission direction, the direction pattern at the transmission direction satisfies the polarization parameters, that is The constraint conditions of the LCMV beamformer can be constructed

[0088]

[0089]

[0090] Step 4, The optimization problem with constraints of the LCMV beamformer can be expressed as

[0091]

[0092]

[0093]

[0094] That is

[0095]

[0096] s.t C H w T = f

[0097] where the constraint matrix constraint response vector I is the identity matrix. is the adjustment direction, represents the steering vector of the desired adjustment component of the conformal array transmission direction pattern. If the adjustment component is the co-polarization direction pattern, then If the adjustment component is the cross-polarization direction pattern, then

[0098] By solving the optimization problem, the transmission weight vector

[0099] w T = R -1 C(C H R -1 C) -1 f

[0100] Next, perform complementary decomposition on the transmission weight vector.

[0101] Step 5, Consider the R -1 component in the transmission weight vector. According to the matrix inversion rule, R -1 can be expressed as

[0102]

[0103] where is an orthogonal projection matrix, and there is

[0104] Step 6. Let Using the matrix inversion rule, decompose C(C H R -1 C) -1 f

[0105]

[0106] The above formula can be further expressed as

[0107]

[0108] Let

[0109]

[0110] Then C(C H R -1 C) -1 f can be expressed as:

[0111]

[0112] Step 7. From Steps 5 and 6, the transmission right vector can be expressed as:

[0113]

[0114] For Simplifying it, we can get:

[0115]

[0116]

[0117]

[0118]

[0119] Let

[0120]

[0121] From the above formula, α′ can be expressed as an equation containing β, that is

[0122]

[0123] Therefore, the transmission right vector can be expressed as

[0124]

[0125]

[0126] It can be sorted and simplified by

[0127]

[0128] to obtain

[0129]

[0130] Let

[0131] w0 = C(C H C) -1 f

[0132]

[0133]

[0134] where P and P ⊥ represent the projection matrix, and there is an equation relationship of P = I - P ⊥ .

[0135] Finally, the transmission right vector can be complementarily decomposed into

[0136] w T = R -1 C(C H R -1 C) -1 f

[0137] = (P ⊥ + βP)w0

[0138] = w1 + βw2

[0139] β is a real number. Once β is determined, the signal power received by the desired dipole at the adjustment direction can be determined. That is, the spatial response of the desired component is controllable. Therefore, the transmission direction pattern can be adjusted by determining the response of the corresponding dipole at each to solve for the final weight vector.

[0140] The following gives how to obtain the adjustment parameter β according to the adjustment direction * .

[0141] Step 8: From w T = (P ⊥ + βP)w0, it can be seen that the transmission right vector w T is updated based on w0 according to β. Therefore, w (k)It can be obtained from the existing weight vector w (k-1) That is P k and are the projection matrices for the k-th iteration and can be obtained according to Step 7

[0142] Step 9: Given the adjustment angle for the k-th iteration Through the weight vector w (k) Adjust the spatial response of the expected adjustment component g (g can be the co-polarization component co or the cross-polarization component x) at to the level value ρ k , which can be expressed as

[0143]

[0144] L (k) is the normalized power for the k-th iteration

[0145] β k can be solved from the above formula according to the expected level ρ k , and there are two feasible solutions

[0146]

[0147]

[0148] where

[0149]

[0150]

[0151]

[0152] w 2,k-1 = P k w (k-1 )

[0153] Step 10: In order to better control the pattern of the expected component, considering that the change in the response of the pattern in the current iteration compared to the pattern in the previous iteration is as small as possible, the change in the pattern between two iterations is measured by the following formula

[0154]

[0155] where

[0156]

[0157] β a and β b The one that makes F(β) take a smaller value is the optimal adjustment parameter β in the current iteration k*, so the weight vector for the k-th iteration is w (k) = w 1,k-1 + β k* w 2,k-1 .

[0158] After multiple iterations, the optimal weight vector w T* , w T* can meet the performance requirements of the transmit beam pattern and achieve transmit beamforming. It should be noted that the adjustment direction for each iteration can be selected as the direction with the largest deviation from the desired beam pattern in the beam pattern obtained from the (k - 1)-th iteration.

[0159] To make the objectives, technical solutions, and technical effects of the present invention clearer, the present invention will be further described in detail through simulation experiments.

[0160] In this experiment, a simulation experiment was conducted on the conformal array transmit beamforming algorithm (LCMV-CD) based on the complementary decomposition of LCMV of the present invention. In the following simulation experiments, the incident signals are all narrowband signals, and the signal wavelength λ = 1. The array is a lower hemisphere array with a radius R = 6λ, the arc length between adjacent rings l = 0.5λ, the interval d c = 0.5λ between adjacent array elements in the same ring, and the total number of array elements is 933. The signal transmission direction The main lobe angle range θ m ∈(θ0 - 7°, θ0 + 7), The depression includes two parts. The angle range 1 is θ i1 ∈(65°, 70°), The angle range 2 is θ i2 ∈(17°, 24°), The methods for comparison include the alternating projection algorithm (AP) and the beamforming algorithm based on convex optimization (CVX) under the same simulation conditions, as well as the initial transmit beam pattern that only performs spatial steering vector complementary addition weighting, that is

[0161] Considering different polarization modes, namely linear polarization and circular polarization: when the polarization mode is linear polarization, considering the main polarization as the horizontal polarization component, that is direction pattern, then the cross polarization is the vertical polarization component, that is direction pattern, the polarization parameters are γ = 0, η = 0, the main polarization parameters γ co = 0, η co = 0, and the cross polarization parameter η x = 0; when the polarization mode is circular polarization, considering the main polarization as left-handed circular polarization, then the cross polarization is right-handed circular polarization, that is, the polarization parameters are The main polarization parameters Cross-polarization parameter

[0162] Simulation Experiment 1: In this simulation, the initial transmit beam pattern with only spatial steering vector complementary addition weighting is simulated. The total transmit beam pattern is as shown in Figure 3 shown The azimuth pattern is as shown in Figure 4 shown The elevation pattern is as shown in Figure 5 shown

[0163] Simulation Experiment 2: In this simulation, considering the polarization mode as linear polarization, the LCMV-CD algorithm, the AP algorithm, and the beamforming algorithm based on convex optimization (CVX) are simulated. The total transmit beam patterns of the three algorithms are as shown in Figure 6 shown, and the top view of the total transmit beam pattern is as shown in Figure 7 shown The azimuth pattern is as shown in Figure 8 shown The elevation pattern is as shown in Figure 9 shown. The co-polarization pattern is as shown in Figure 10 shown. The cross-polarization pattern is as shown in Figure 11 shown. The elevation plane profile pattern is as shown in Figure 12 shown. The azimuth plane profile pattern is as shown in Figure 13 shown. The performance of each algorithm is shown in the following table:

[0164]

[0165] It should be noted that in linear polarization, considering the co-polarization as horizontal polarization and the cross-polarization as vertical polarization. Therefore, as shown in it can be considered that the polarization mode with the main beam pointing down satisfies the polarization parameters

[0166] Simulation Experiment 3: In this simulation, considering the polarization mode as circular polarization, the LCMV-DP algorithm, the AP algorithm, and the beamforming algorithm based on convex optimization are simulated. The total transmit beam patterns of the three algorithms are as shown in Figure 14 shown, and the top view of the total transmit beam pattern is as shown in Figure 15 shown The azimuth pattern is as shown in Figure 16 shown The

[0167] elevation pattern is as shown in Figure 17 shown. The co-polarization pattern is as shown in Figure 18 shown. The cross-polarization pattern is as shown in Figure 19 shown. The elevation plane profile pattern is as shown in Figure 20 shown. The azimuth plane profile pattern is as shown in Figure 21 shown. The performance of each algorithm is shown in the following table:

[0168]

[0169] As can be seen from the above simulation experiments, the method of the present invention solves the problem that beamforming cannot be performed on a conformal array by designing a weight vector so that the performance of each item of the transmitted beam pattern meets the requirements. The method of the present invention can effectively reduce the peak sidelobe level and cross-polarization level while satisfying that the main beam pointing does not shift, and form a depression within the desired angle range. At the same time, the method of the present invention can effectively realize the constraint on the polarization mode, and finally the polarization parameters are close to the desired polarization parameters. Compared with the classical conformal array transmitted beamforming algorithm, the performance of the method of the present invention is overall better than the alternating projection algorithm. The alternating projection algorithm will cause the shift of the main beam pointing and slightly worse polarization performance. The performance of the method of the present invention is similar to that of the beamforming algorithm based on convex optimization. However, the beamforming algorithm based on convex optimization does not consider the influence of cross-polarization on practical applications and cannot realize the constraint on cross-polarization. In addition, under the same simulation conditions, compared with the beamforming algorithm based on convex optimization, the method of the present invention can obtain the optimal weight vector more quickly and takes less time.

[0170] The above is only the specific implementation manner of the present invention. Any feature disclosed in this specification, unless specifically described, can be replaced by other equivalent or similar-purpose alternative features; all the features disclosed, or all the steps in any method or process, except for mutually exclusive features and / or steps, can be combined in any way; any non-essential addition or replacement made by those skilled in the art according to the technical features of the technical solution of the present invention belongs to the protection scope of the present invention.

Claims

1. A conformal array transmitting beamforming method based on LCMV complementary decomposition, characterized in that The total number of array elements of the conformal array in the rectangular coordinate system is N. The array elements use rectangular microstrip antennas and are placed at horizontal and vertical intervals. The nth array element has an element pattern in the azimuth angle and the elevation angle The element pattern in the direction is where θ is the azimuth angle variable, is the elevation angle variable; the spatial steering vector of the conformal array is In and The steering vectors in the directions are respectively and The specific steps of transmit beamforming are as follows: Step 1) Construct a beamformer of linearly constrained minimum variance (LCMV), and set the weight vector w T = R -1 C (C H R -1 C) -1 f, where C is the constraint matrix, f is the constraint response vector, and R is the spatial correlation matrix, H is the conjugate transpose; Step 2) Complementary decompose the weight vector into two components w1 and w2, which are the results weighted by adjusting the parameter β: w T = R -1 C(C H R -1 C) -1 f = w1 + βw2; The components w1 and w2 are: Among them, To adjust the beam direction, w (0) is the LCMV weight vector w T The initial iteration value of w (0) = C(C H C) -1 f, P ⊥ And P is the projection matrix; Among them, represents the steering vector of the conformal array radiation pattern adjustment point, which can be adjusted to make set as the main polarization pattern or the cross polarization pattern. The intermediate matrix and the intermediate parameter α are respectively: where I is the identity matrix, and p 2 represents the power of the emission pattern adjustment point; Step 3) Perform the k-th iteration: Determine the k-th adjustment direction Obtain the k-th expected level value ρ k Expression: using the weight vector w obtained in the previous iteration (k-1) in the component w 1,k-1 and w 2,k-1 and the preset expected level value ρ k , two candidate results β of the adjustment parameter β are obtained a and β b : Among them, is to take the real part, B(1, 2) means taking the element in the first row and second column of matrix B, and B(2, 2) means taking the element in the second row and second column of matrix B; Intermediate quantity Step 4) Take β a and β b as the adjustment parameters β respectively and input them into the function F(β) for calculation. Select the one that makes the value of F(β) smaller as the optimal solution β k* obtained in the k-th iteration, so as to determine the weight vector w (k) , w (k) = w 1,k-1 + β k* w 2,k-1 ; where, ||·||2 is the second norm; Step 5) Determine whether the iteration stop condition is reached. If so, use the weight vector w obtained in the k-th iteration (k) as the optimal transmission weight vector w T* for conformal array transmit beamforming; otherwise, update k = k + 1 and return to Step 3); the iteration stop condition is that the transmit pattern determined by the weight vector w (k) satisfies the constraint or k reaches the maximum value.

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