An array antenna shaped beam method for minimizing the main and sidelobe spacing

Abstract

The present invention provides a method for shaping the beam of an array antenna by minimizing the main and sidelobe spacing. Aiming at the problem of optimizing the width of the transition band region, the present invention directly takes the width of the transition band as the object of relaxation constraint, introduces a relaxation vector and a penalty vector in the relaxation region, and modifies the objective function by using the penalty vector, so as to obtain a convex and relaxed optimized beam shaping problem; then uses an existing method to obtain a reference beam shaping; finally, uses an iterative method to continuously narrow the width of the transition band, and at the same time realizes a beam shaping with low sidelobes and narrow main lobe ripple.

Description

Technical Field

[0001] The present invention relates to the beamforming technology of array antennas, and particularly to a method for shaping the beam of an array antenna that minimizes the main-to-side lobe spacing. Background Art

[0002] Array antennas have good beamforming capabilities, such as the pencil beam pattern (PBP) and the shaped beam pattern (SBP), and thus have a wide range of applications in the fields of communication, radar, remote sensing, etc. A typical application scenario of SBP is the reception of satellite signals by a mobile platform. Generally speaking, SBP has the following requirements: in order to reduce the attenuation of signals during reception, a narrow main lobe ripple (MRL) is required, and at the same time, in order to combat the noise and clutter interference of the external environment, a low sidelobe level characteristic is needed. To meet the above requirements, the existing method is to optimize the excitation of each array element to synthesize a wide beam waveform with a specific shape, and this kind of method is usually called the shaped beam pattern synthesis method (SBPS).

[0003] When the existing method synthesizes the shaped beam, it is necessary to pre-divide the main lobe region and the side lobe region in the entire array synthesis region according to experience. Here, the concept of "transition band" is introduced from the filter field, and the region between the main lobe region and the side lobe region is denoted as the transition band region.

[0004] The existing method indirectly determines the transition band region through the first two regions, that is, the width of the transition band is set according to experience, which will result in the width of the transition band region not being optimal.

[0005] An inappropriate transition band setting will affect the performance of the shaped beam. For example, in order to achieve narrow ripple and low sidelobes, a wide transition band is set, which will cause energy leakage from the transition band region; although a narrow transition band region can reduce energy leakage, it will make it difficult to achieve narrow main lobe ripple or low sidelobes.

[0006] Consider a linear array antenna with N elements, and let its position be (r n ), n = 1,..., N. The electric field intensity of the array antenna is described as:

[0007]

[0008] where ω n and a n (θ) are respectively the excitation and the array factor of the nth element factor. En The electric field strength representing the array element factor of the nth one, θ represents the direction, and θ varies within and denote the array synthesis direction change region as Θ.

[0009] Performing vectorization on the above formula, we can obtain:

[0010] E(θ) = w H a(θ) (2)

[0011] where

[0012]

[0013] where w is the excitation vector of the array antenna, a represents the intensity of the array antenna, is the spatial wave number of the electromagnetic wave, r n represents the position of the nth array element, and H represents the conjugate transpose of the matrix.

[0014] Under normal circumstances, the array element characteristics of the array antenna are the same. When modeling, it can be assumed that it has an omnidirectional radiation characteristic. At this time, a(θ) = [a 1 (θ), …, a N (θ)] H is a coefficient related only to the array factor.

[0015] A typical convex optimization method for array excitation to solve the SBPS problem is to solve the following optimization problem:

[0016] min w，η η

[0017] s.t. max ||w H a(θ m ) 2 - d(θ m )| ≤ η, θ m ∈ Θ ML (4)

[0018] |w H a(θ s )| ≤ ρ, θ s ∈ Θ SL

[0019] where η is the maximum main lobe ripple level to be obtained, and ρ is the preset sidelobe level. θ m and θ s respectively represent the main lobe direction and the sidelobe direction. Θ ML and Θ SL respectively represent the main lobe region and the sidelobe region, and d is the shape of the desired main lobe beam of the array. Here, the region between Θ ML and Θ SL is denoted as ΘTB 。 Summary of the Invention

[0020] The technical problem to be solved by the present invention is to complete the method of shaping the beam of an array antenna by optimizing to obtain the minimum transition band width between the main lobe and the side lobe.

[0021] Convex optimization is widely used in array synthesis problems because it can handle any array element configuration and fully consider array element coupling. In addition, by representing the problem in the form of a convex problem, the optimal solution can be easily obtained through an interior point optimization tool. When facing non-convex array synthesis problems, the problem can also be transformed into a convex problem form by introducing methods such as relaxation, approximation, and iteration. Generally, the SBPS problem is a non-convex problem, and the present invention also needs to transform the SBPS problem into a convex problem.

[0022] The technical solution adopted by the present invention to solve the above technical problems is an array antenna beam shaping method for minimizing the main lobe and side lobe spacing, including the following steps:

[0023] 1) Setting step:

[0024] 1-1) Set the reference excitation w 0 , main lobe region Θ M , side lobe region Θ S , transition band Θ TB , steering vector a(θ), the maximum side lobe level ρ, and the difference value δ between the main lobe ripple formed by the excitation w of the array antenna and the main lobe ripple formed by the reference excitation w, where θ represents the direction; 0 between the main lobe ripples formed.

[0025] 1-2) Discretize the set transition band Θ TB to obtain the length L of the transition band Θ TB ;

[0026] 1-3) Calculate the matrix F according to the length L:

[0027]

[0028] where the element variable variable

[0029] 1-4) Calculate the penalty vector v. The penalty vector v is L-dimensional, and l is a variable from 1 to L. The l-th element in v is expressed as:

[0030]

[0031] 1-5) Initialize the reference vector w with the excitation obtained by using the existing convex optimization method for solving the array excitation of the SBPS problem 0;

[0032] 2) Iterative steps:

[0033] 2-1) Set the length of t in the r-th iteration of the relaxation vector t to Len r (t), and set Len 0 (t) = L + 1; then initialize the iteration number r = 1, Len 1 (t) = L;

[0034] 2-2) Determine whether Len r-1 (t) > Len r (t) is satisfied. If so, sequentially execute steps 2-3) to 2-6); otherwise, execute 3) the output step;

[0035] 2-3) Obtain the excitation w of the array antenna and the relaxation vector t in the current iteration process through a convex optimization problem:

[0036] min w，t vt

[0037] s.t. |w 0 H a(θ m )a(θ m ) H (w - w 0 )| ≤ δ, (θ m ) ∈ Θ ML

[0038] w H a(θ l )| ≤ ρ + t l , (θ l ) ∈ Θ TB

[0039] t ≥ 0

[0040] Ft ≤ 0

[0041] w H a(θ s )| ≤ ρ, (θ s ) ∈ Θ SL

[0042] Among them, θ m and θ s represent the main lobe direction and the sidelobe direction respectively, t l is the l-th element in the relaxation vector t, H represents the conjugate transpose of the matrix;

[0043] 2-4) Remove the elements close to 0 in the relaxation vector t to update t; the elements close to 0 are the elements less than the preset value;

[0044] 2 - 5) Update the length L to the length of the relaxed vector t after removing elements close to 0, and then update the matrix F according to the updated L in the manner of steps 1 - 3); Divide 1 by each element in the updated relaxed vector t to obtain a vector Then use to assign values to the penalty vector v to complete the update of the penalty vector v;

[0045] 2 - 5) After updating the iteration number r = r + 1, return to step 2 - 2);

[0046] 3) Output step: Output the excitation w of the current array antenna to complete the beamforming of the array antenna. The beneficial effect of the present invention is that, aiming at the problem of optimizing the width of the transition band region, the present invention directly uses the width of the transition band as the object of relaxation constraint, introduces a relaxation vector and a penalty vector in the relaxation region, and modifies the objective function by using the penalty vector, so as to obtain a convex and relaxation - optimized beamforming problem; Then use the existing method to obtain a reference beamforming; Finally, use the iterative method to continuously narrow the width of the transition band while realizing a shaped beam with low sidelobes and narrow main - lobe ripple. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 It is a schematic diagram of the main - lobe region, sidelobe region and the transition - band region therebetween in a flat - top beam;

[0048] Figure 2 It is a schematic of the performance of the shaped flat - top beam; 'Ref.' represents the beam pattern of the reference excitation; '1st' to '4th' are the processes of four - time iteration adaptive minimization of the transition band of the flat - top beam; Figure 2 (a) shows the process of minimizing the transition band, Figure 2 (b) shows the change of the main - lobe ripple, Figure 2 (c) shows the change of the penalty vector t during iteration.

[0049] Figure 3 It is the beam pattern of the flat - top beam under different gap coefficients δ; Figure 3 (a), Figure 3 (b), Figure 3 (c), Figure 3 (d) respectively show the processes of adaptive minimization of the transition band of the flat - top beam when the gap coefficients δ are 0.0005, 0.0010, 0.0020, 0.0025 respectively.

[0050] Figure 4 It is the beam pattern of the flat - top beam under different sidelobe - level constraint values ρ; Figure 4 (a), Figure 4 (b), Figure 4(c) respectively show the process of the flat-top beam continuously shrinking the transition band under different sidelobe level constraints ρ = -30 dB, -40 dB, and -50 dB.

[0051] Figure 5 Among them are the flat-top beam patterns under different main lobe widths;

[0052] Figure 5 (a), Figure 5 (b), Figure 5 (c), Figure 5 (d) show the process of the flat-top beam adaptively minimizing the transition band when the main lobe widths are [-5°, 5°], [-10°, 10°], [-15°, 15°], and [-25°, 25°] respectively.

[0053] Figure 6 Among them are the flat-top beam patterns under different beam center points; Figure 6 (a), Figure 6 (b), Figure 6 (c), Figure 6 (d) show the process of the flat-top beam adaptively minimizing the transition band when the different beam center points are 10°, 20°, 30°, and 50° respectively. Specific implementation manner

[0054] Taking the minimization of the transition band Θ TB as the optimization objective, then the SBPS problem can be rewritten as:

[0055]

[0056] where, min|Θ TB | represents minimizing the width of the transition band, C A B represents the complement set, which is the set in set A except set B, A ∪ B represents the union of A and B, C Θ (Θ ML ∪Θ SL ) indicates that the transition band region is the region in the direction change range Θ except the main lobe region Θ ML and the sidelobe region Θ SL . Here, the slack vector t is introduced. The dimension of the slack vector t represents the width of the transition band region, and the length is denoted as L. Add the constraint on the transition band region to Equation (5) and rewrite Equation (5) as:

[0057]

[0058] In the formula, ||·|| 0 is the l 0 norm, whose function is to count the number of non-zero elements in the slack vector t, l = 1,..., L. When the slack vector t has tl When = 0, it can be seen that the transition band constraint and the sidelobe constraint are equivalent. At this time, the transition band region becomes smaller, while the sidelobe region expands accordingly. This is the principle of reducing the transition band. However, such a form cannot guarantee the effectiveness of the transition band constraint. The applicant adds two more constraint conditions to strongly constrain the relaxation vector t. Here, a matrix F of size (L - 2)×L and a matrix Q of size are introduced:

[0059]

[0060]

[0061] where the variable the variable

[0062] Adding the matrix F and Q to constrain the relaxation vector t, then problem (6) can be written as:

[0063] min w，t ||t|| 0

[0064] s.t.max||w H a(θ m )| 2 -d(θ m )|≤η,(θ m )∈Θ ML

[0065] |w H a(θ l )|≤ρ+t l ,(θ l )∈Θ TB (9)

[0066] t≥0

[0067] Ft≤0

[0068] |w H a(θ s )|≤ρ,(θ s )∈Θ SL

[0069] Since the above problem has a non - convex objective function ||t|| 0 and the main lobe constraint condition max||w H a| 2 -d(θ m )|≤η, in order to transform the above problem into a convex problem, the objective function and the main lobe constraint condition are relaxed. Here, an L - dimensional penalty vector v is introduced, and the l - th element in v is expressed as:

[0070]

[0071] and use the reference excitation w obtained by the conventional solution formula (4) 0 to replace the desired waveform d. Since the max|·| function represents that the maximum value of all internal values is less than the constraint value, as long as it is ensured that all values of |·| are less than the constraint value, the two can be equivalent. Therefore, Equation (9) can be rewritten as:

[0072] min w，t vt

[0073]

[0074] where δ represents the difference between the main lobe ripple formed by the to-be-solved excitation w and the main lobe ripple formed by the reference excitation w 0

[0075] Note that solving the problem (11) once will form a new L-dimensional relaxation vector t. At this time, updating the matrix F and the penalty vector v may further narrow the width of the transition band. Therefore, the present invention adopts an iterative method to minimize the width of the transition band region.

[0076] The specific implementation steps are as follows:

[0077] Step 1: Set the main lobe region Θ M , the sidelobe region Θ S , the transition band region Θ TB , the steering vector a(θ), the maximum sidelobe level ρ(θ), and the difference value δ;

[0078] Step 2: Calculate the length L of Θ TB ;

[0079] Step 3: Initialize the matrix F using formulas (7) and (8) and initialize the penalty vector v using formula (10);

[0080] Step 4: Obtain the reference vector w 0 ;

[0081] Step 5: Define the length of t in the r-th iteration as Len r (t);

[0082] Step 6: Initialize r = 1, Len 0 (t) = L + 1, Len 1 (t) = L;

[0083] Step 7: If Len r-1 (t) > Len r (t), then perform steps (8)-(11); if not satisfied, perform step 12; ​

[0084] Step 8: Obtain {wt} by solving problem (11);

[0085] Step 9: Update the slack vector t by removing the elements close to 0 in the slack vector t, specifically, t l ≤10 -4 ;

[0086] Step 10: Recalculate the length L of the vector t, and then update the matrix F according to formulas (7) and (8), and use to update the penalty vector v;

[0087] Step 11: r = r + 1, return to Step 7;

[0088] Step 12: Output w.

[0089] Experimental effect

[0090] Experiment 1: A half-wavelength uniformly distributed linear array with 22 elements is adopted. The present invention uses the excitation obtained by solving the SBPS problem by the traditional method as the reference excitation, and demonstrates the effect of the present invention in shrinking the width of the transition band in the flat-topped beam.

[0091] Parameter configuration: Main lobe region Θ ML = [-20°, 20°], transition band region Θ TB = [-40°, -20°] ∪ [20°, 40°], sidelobe region Θ SL = [-90°, -20°] ∪ [20°, 90°]; Set the gap coefficient δ = 0.0015, the sidelobe level constraint value ρ = -30dB, and the angular resolution is 0.1°.

[0092] For the detailed data of the MRL and the transition band region, Table 1 shows the MRL and the transition band Θ TB obtained in each of the four iterations:

[0093] # MRL (dB) <![CDATA[Θ TB (°)]]> Ref. ±0.013 dB [-30°,-20°]∪[20°,30°] 1st ±0.124 dB [-34.1°,-20°]∪[20°,34.1°] 2nd ±0.088 dB [-30.0°,-20°]∪[20°,30.0°] 3rd ±0.144 dB [-29.2°,-20°]∪[20°,29.2°] 4th ±0.202 dB [-29.0°,-20°]∪[20°,29.0°]

[0094] Table 1

[0095] Figure 2 (a) and Θ in Table 1 TB show that compared with the initially set transition band region [-40°, -20°] ∪ [20°, 40°], after four iterations of solution, the transition band region is [-29.0°, -20°] ∪ [20°, 29.0°], and the transition bands on both sides of the main lobe are reduced by 11° respectively. Figure 2 (b) and the MRL in Table 1 show that compared with the ripple of the reference beam, the ripple fluctuation increases by about ±0.2dB, which is an acceptable change in practical applications. Figure 2(c) indicates that within four iterations, the near-zero region (≤ 10 -4 ) of the penalty vector t expands with iteration, while the non-zero region decreases, indicating a corresponding reduction in the width of the transition band.

[0096] The above results show that the method of the present invention can adaptively minimize the width of the transition band and obtain a narrow main lobe ripple MRL and a flat-topped beam with ideal side lobes.

[0097] Experiment 2: A half-wavelength uniformly distributed linear array with 22 elements is used. The present invention uses the excitation obtained by solving the SBPS problem by the traditional method as the reference excitation and demonstrates the influence of the gap coefficient δ on the main lobe ripple MRL and the transition band width Θ TB of.

[0098] Parameter configuration: The main lobe region Θ ML = [-20°, 20°], the transition band region Θ TB = [-40°, -20°] ∪ [20°, 40°], the side lobe region Θ SL = [-90°, -20°] ∪ [20°, 90°];

[0099] Set the side lobe level constraint value ρ = -30 dB and the angular resolution to 0.1°. The gap coefficients are set to 0.0005, 0.0010, 0.0020, and 0.0025 respectively.

[0100] Table 2 shows the MRL and the transition band Θ TB obtained under different gap coefficients δ:

[0101] δ MRL (dB) <![CDATA[Θ TB (°)]]> 0.0005 ±0.101 dB [-29.4°,-20°]∪[20°,29.4°] 0.0010 ±0.167 dB [-29.2°,-20°]∪[20°,29.2°] 0.0020 ±0.238 dB [-28.8°,-20°]∪[20°,28.8°] 0.0025 ±0.346 dB [-28.7°,-20°]∪[20°,28.7°]

[0102] Table 2

[0103] From Figure 3 it can be observed that when the gap coefficients are different, both the reduction of the transition band and the fluctuation range of the MRL will change. More detailed data are provided in Table 2, which shows that when the gap coefficient δ ranges from 0.0005 to 0.0025, the obtained MRL correspondingly increases from ±0.101 dB to ±0.346 dB, and the obtained transition band width correspondingly shrinks by 0.7°. This means that under the same other parameters, the larger the gap coefficient δ, the larger the main lobe ripple obtained, but the relatively narrower the obtained transition band width. This also indicates that the main lobe ripple and the transition band width are a pair of performance parameters that affect each other. In practical applications, the gap coefficient δ needs to be comprehensively selected according to the required MRL and the range of the transition band.

[0104] Experiment 3: A half-wavelength uniformly distributed linear array with 22 elements is used. The present invention takes the excitation obtained by solving the SBPS problem using the traditional method as the reference excitation, and demonstrates the influence of the sidelobe level constraint value ρ on the main lobe ripple MRL and the transition band width.

[0105] Parameter configuration: Main lobe region Θ ML = [-20°, 20°], transition band region Θ TB = [-40°, -20°] ∪ [20°, 40°], sidelobe region Θ SL = [-90°, -20°] ∪ [20°, 90°];

[0106] Set different sidelobe level constraint values ρ = -30dB, -40dB, -50dB, and the angular resolution is 0.1°.

[0107] MRL and transition band Θ obtained under different sidelobe level constraint values ρ TB As shown in Table 3:

[0108] ρ (dB) MRL (dB) <![CDATA[Θ TB (°)]]> -30 dB ±0.142 dB [-29.3°,-20°]∪[20°,29.3°] -40 dB ±0.300 dB [-29.4°,-20°]∪[20°,29.4°] -50 dB ±0.443 dB [-29.7°,-20°]∪[20°,29.7°]

[0109] Table 3

[0110] For an antenna array, a low sidelobe means strong anti-interference ability. Figure 4 It can be seen that this experiment demonstrates the performance of the present invention under different sidelobe constraint values, that is, in the case of sidelobe level constraint values ρ = -30dB, -40dB, -50dB, the process of continuously narrowing the transition band. It can be observed that the obtained MRL and transition band are different in each case. More detailed data are shown in Table 3. It can be seen that when ρ decreases from -30dB to -50dB, the main lobe ripple MRL increases from ±0.142dB to ±0.443dB, an increase of about ±0.3dB, and the corresponding obtained transition band width is reduced by about 0.4°. This experiment shows that the sidelobe level constraint value ρ and the main lobe ripple MRL are a pair of interacting parameters, and the lower the sidelobe, the wider the MRL.

[0111] Experiment 4: A half-wavelength uniformly distributed linear array with 22 elements is used. The present invention takes the excitation obtained by solving the SBPS problem using the traditional method as the reference excitation, and demonstrates the influence of the main lobe width on the main lobe ripple MRL and the transition band width.

[0112] Parameter configuration: Main lobe region Θ ML Are respectively set to [-5°, 5°], [-10°, 10°], [-15°, 15°], [-25°, 25°];

[0113] The width of the transition zone on both sides of the main lobe is 20°, and the remaining area is the sidelobe region; the sidelobe level constraint value ρ is set to -30 dB, and the angular resolution is 0.1°.

[0114] Different main lobe region widths Θ ML The obtained MRL and transition zone Θ obtained below TB As shown in Table 4:

[0115] ΘML (°) MRL (dB) <![CDATA[Θ TB (°) <!-- 7 -->]]> [-5°,5°] ±0.082 dB [-14.5°,-5°]∪[5°,14.5°] [-10°,10°] ±0.088 dB [-19.6°,-10°]∪[10°,19.6°] [-15°,15°] ±0.092 dB [-24.9°,-15°]∪[15°,24.9°] [-25°,25°] ±0.077 dB [-34.8°,-25°]∪[25°,34.8°]

[0116] Table 4

[0117] As Figure 5 shown is the process of minimizing the transition zone adaptively for a flat-top beam when the main lobe widths are [-5°, 5°], [-10°, 10°], [-15°, 15°], and [-25°, 25°] respectively. Table 4 shows the data on the obtained MRL and transition zone. When the main lobe width increases from [-5°, 5°] to [-15°, 15°], the main lobe ripple increases from ±0.082 dB to ±0.092 dB, an increase of about ±0.01 dB, and the obtained transition zone expands by about 0.4°. However, when the main lobe region is [-25°, 25°], it does not increase further but decreases, because the ripple of the reference excitation gradually increases with the change of the main lobe width. When the ripple of the reference excitation main lobe is greater than a certain threshold, the present invention optimizes and reduces the ripple. Another advantage of the present invention is that the obtained transition zone width is smaller than that of the reference beam.

[0118] Experiment 5: Using a half-wavelength uniformly distributed linear array with 22 elements, the present invention uses the excitation obtained by solving the SBPS problem by the traditional method as the reference excitation and demonstrates the beam scanning ability of the present invention.

[0119] Parameter configuration: The width of the main lobe is set to 20°, and the beam center point θ c is 10°, 15°, 20°, and 25° respectively. The width of the transition zone on both sides of the main lobe in the beam is 20°, and the remaining area is the sidelobe region; the sidelobe level constraint value ρ is set to -30 dB, and the angular resolution is 0.1°.

[0120] Different beam center points θ c The obtained MRL and transition zone Θ obtained below TB As shown in Table 5 below:

[0121] <![CDATA[Θ c (°)]]> MRL (dB) <![CDATA[Θ TB (°)]]> 10° ±0.135 dB [-18.8°,-10°]∪[30°,39.5°] 15° ±0.238 dB [-14.0°,-5°]∪[35°,44.7°] 20° ±0.272 dB [-9.20°,-0.0°]∪[40°,49.3°] 25° ±0.406 dB [-3.00°,5.0°]∪[45°,54.8°]

[0122] Table 5

[0123] Figure 6It shows that the present invention can achieve the transition band with minimized flat-topped beam at different beam center points, and at the same time achieve narrow ripple and low sidelobes. And as the distance from 0° increases, the main lobe ripple MRL deteriorates accordingly. Table 5 shows the obtained MRL and transition band region data. When the beam center point θ c ranges from 10° to 25°, the obtained MRL ranges from ±0.135 dB to ±0.406 dB. In order to obtain a narrow main lobe ripple, the beam center point θ c needs to be close to 0°. Taking the beam center point θ c = 10° as an example, the width of the left transition band is 8.8°, and the width of the right transition band is 9.5°, that is, the left side is 0.7° narrower than the right side. For the flat-topped beam, the greater the beam deviation from 0°, the greater the obtained MRL.

Claims

1. An array antenna shaped beam method for minimizing the main and side lobe spacing, characterized in that, it includes the following steps: 1) Setting step: 1-1) Set the reference excitation w 0 , main lobe region Θ M , sidelobe region Θ S , transition band Θ TB , the main lobe ripple formed by the steering vector a(θ), the maximum sidelobe level ρ, and the excitation w of the array antenna, and the reference excitation w 0 , the gap value δ between the main lobe ripple formed, where θ represents the direction; 1-2) Discretize the set transition zone Θ TB to obtain the length L of the transition zone Θ TB ; 1-3) Calculate matrix F according to length L: Among them, the element variable variable 1-4) Calculate penalty vector v. Penalty vector v is L-dimensional, l is a variable from 1 to L, and the l-th element in v is expressed as: 1 - 5) Initialize the reference vector w with the excitation obtained by using the existing convex optimization method for array excitation to solve the SBPS problem 0 ; 2) Iterative step: 2-1) Set the length of the relaxation vector t in the r-th iteration to be Len r (t), and set Len 0 (t) = L + 1; then initialize the iteration number r = 1, Len 1 (t) = L; 2-2) Determine whether Len r-1 (t) > Len r (t) is satisfied. If so, sequentially execute steps 2-3) to 2-6). Otherwise, execute step 3) to output the step; 2-3) Obtain the excitation w of the array antenna and the relaxation vector t in the current iteration process through a convex optimization problem: min w，t vt w H a(θ l ) ≤ ρ + t l , (θ l ) ∈ Θ TB t≥0 Ft≤0 w H a(θ s )≤ρ,(θ s )∈Θ SL where, θ m and θ s represent the main lobe direction and the side lobe direction respectively, t l is the l-th element in the relaxation vector t, H denotes the conjugate transpose of the matrix; 2-4) Update t by removing the elements close to 0 in the relaxation vector t; the elements close to 0 are the elements less than the preset value; (2-5) Update the length L to the length of the slack vector t after removing elements close to 0, and then update the matrix F according to the updated L in the manner of steps (1-3); divide each element in the updated slack vector t by 1 to obtain a vector Then use to assign values to the penalty vector v to complete the update of the penalty vector v; 2-6) After updating the iteration number r = r + 1, return to step 2-2); 3) Output step: Output the current excitation w of the array antenna to complete the beam shaping of the array antenna.

2. The method according to claim 1, characterized in that, The preset value is 10 -4 .

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