Array error robust narrowest beam synthesis method based on sparse optimization

Abstract

The invention provides an array error robust narrowest beam synthesis method based on sparse optimization. According to the method, for the problem that a main lobe area of an array antenna cannot be accurately determined under the low side lobe constraint, slack variables are introduced, the high-gain spot beam optimization problem is converted into the main lobe width minimization problem, robustness control over array amplitude-phase errors is introduced into the constraint, and the beam forming performance under the condition that array errors exist is remarkably improved. The method comprises the following calculation steps: 1, constructing a comprehensive convex constraint condition of a directional diagram according to beam pointing and an upper bound of an array amplitude-phase error; 2, constructing a cost function based on a logarithm minimization problem of an e index sum, and constructing a directional diagram comprehensive convex optimization problem; and 3, solving an optimization problem by using a convex optimization solving tool. According to the method, while the width of the main lobe is minimized, accurate control over the characteristics of the main lobe and the side lobe of the directional diagram is achieved under the condition that the amplitude-phase error boundary is given.

Description

Technical Field

[0001] The present application relates to the technical field of digital array antennas, and in particular to an array error robust narrowest beam synthesis method based on sparse optimization. Background Art

[0002] In the field of digital array beam pattern synthesis, the existing pattern synthesis algorithms are difficult to maximize the main lobe gain under arbitrary sidelobe control, and do not consider robustness optimization for array errors on this basis. For example, the classic Chebyshev synthesis method achieves uniform sidelobes by adding windows and makes the main lobe width of the beam reach the first zero point, while uniformly controlling the sidelobes to a given level, and cannot achieve control of arbitrary sidelobe shapes. At the same time, the Chebyshev synthesis method is only applicable to arrays composed of uniformly arranged and isotropic unit antennas. The convex optimization synthesis method can handle optimization problems with multiple constrained objectives, such as sidelobe level control, main lobe width adjustment, array gain optimization, and system robustness. By making reasonable compromises among multiple performance indicators, optimized comprehensive performance is achieved. The comprehensive optimization also has no restrictions on the characteristics of the array structure and the unit antenna pattern. The typical convex optimization problem of a high-gain main lobe and low-sidelobe pattern is expressed as:

[0003]

[0004] stF(θ)≥M U θ∈Ω M

[0005] F(θ)≤S U θ∈Ω S

[0006] Among them, F(θ) and F d (θ) is the array amplitude pattern and the expected amplitude pattern at angle θ, M U is the lower bound of the amplitude constraint in the main lobe area of ​​the pattern, S U is the upper bound of the amplitude constraint in the sidelobe area of ​​the pattern.

[0007] With the increasing demand for pattern characteristics, the beam patterns formed using convex optimization techniques are increasingly susceptible to non-ideal factors. These factors will cause amplitude and phase errors in the array. If these errors cannot be effectively controlled, the beam patterns obtained by convex optimization will be distorted, thus affecting the performance of the array system.

[0008] Therefore, reducing the impact of these amplitude and phase errors is critical to achieving high-quality beam patterns. Summary of the invention

[0009] The present application provides an array error robust narrowest beam synthesis method based on sparse optimization, which can be used to solve the technical problem of large amplitude and phase errors in the prior art. The method provided in the present application aims to ensure that the narrowest main lobe beam is generated at a specified angle while significantly reducing the impact of array amplitude and phase errors on the comprehensive performance of the pattern. This technology has the characteristics of robust control of pattern distortion in the presence of array amplitude and phase errors, ensuring the shaping performance of the point beam pattern. The method has low computational complexity and can be widely used in digital array-based radar, communication, sonar, radio astronomy, and voice signal processing systems to achieve the requirement of robust control of the main lobe and side lobes of the antenna pattern in the presence of certain array errors.

[0010] The present application provides an array error robust narrowest beam synthesis method based on sparse optimization, the method comprising:

[0011] Step 1, determine the pattern power constraint according to the beam pointing and the expected sidelobe level distribution, introduce slack variables and construct convex constraint conditions for pattern synthesis;

[0012] Step 2, construct a cost function based on the logarithm minimization problem of the sum of the e-exponentials of the slack variables, and build a convex optimization problem for directional pattern synthesis;

[0013] Step 3: Use the convex optimization solving tool to solve the convex optimization problem of pattern synthesis.

[0014] Compared with the prior art, the present invention has the following significant advantages:

[0015] (1) The present invention introduces robust control of array amplitude and phase errors in the narrowest beam optimization problem, which can effectively reduce the impact of array amplitude and phase errors on the pattern optimization results while ensuring the narrowest beam main lobe width and sidelobe control;

[0016] (2) The present invention improves the complexity of the solution of the present invention by converting the main lobe amplitude constraint in the beam synthesis convex optimization problem into a real part constraint, and converting the non-convex constraint into a convex constraint without changing the constraint effect.

[0017] (3) The method proposed in the present invention can obtain the optimization result through only one convex optimization solution, and can obtain the global optimal solution of the optimization problem. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1 It is a flow chart for realizing the method of the present invention.

[0019] Figure 2 This is the optimized directional pattern of the uniform linear array with N=20 array elements in this example using the method of the present invention in the presence of array amplitude and phase errors.

[0020] FIG3 is a comparison between the directional diagram obtained by the optimization method of the present invention and the non-robust directional diagram under the conditions of no error and error with similar sidelobe performance in this example. DETAILED DESCRIPTION

[0021] In order to make the objectives, technical solutions and advantages of the present application clearer, the implementation methods of the present application will be further described in detail below with reference to the accompanying drawings.

[0022] The following first introduces the embodiments of the present application in conjunction with the accompanying drawings.

[0023] Combination Figure 1 , an array error robust narrowest beam synthesis method based on sparse optimization, comprising the following steps:

[0024] Step 1: Determine the pattern power constraint based on the beam pointing and the expected sidelobe level distribution, introduce slack variables and construct convex constraints for pattern synthesis.

[0025] Step 2, construct a cost function based on the logarithm minimization problem of the sum of the e-exponentials of the slack variables and build a convex optimization problem for directional pattern synthesis.

[0026] Step 3: Use the convex optimization solving tool to solve the convex optimization problem of pattern synthesis.

[0027] The specific implementation process of step 1 is further given:

[0028] Step 1-1, determine the desired pattern F according to the beam pointing to the sidelobe level distribution d (θ);

[0029] Assume that the desired beam direction of the pattern is θ p , where θ p is the angle predetermined by the project indicator; the ideal expectation of spot beamforming is that the directional pattern in other angle areas except the pointing angle must meet the low sidelobe power constraint; therefore, the ideal expected directional pattern is expressed as Among them, M u is the expected amplitude of the main lobe, S u is the expected amplitude of the side lobe, Ω all is the angle area of ​​the entire integrated pattern, where \ represents except, that is, except Ω all Except for θ p Outside angle area.

[0030] Step 1-2, introduce the slack variable ρ(θ) and modify the desired pattern to F d2 (θ), construct the monotonicity constraint of the directional graph.

[0031] Since the beam pattern cannot meet the ideal expectation, the pointing angle θ pThe slack variable ρ(θ) is introduced in the nearby area to relax the power constraint of the pattern in the corresponding angle area; and when ρ(θ) = 0, θ is included in the sidelobe area; the modified desired pattern is expressed as: Among them, Ω c is the angular area near the beam pointing, Ω s is the side lobe angle area;

[0032] Next, we use the monotonicity of the main lobe area of ​​the pattern to impose constraints on ρ(θ); since θ p The directional pattern level in the left angle area is monotonically increasing, that is, ρ(θ1)<ρ(θ2), where θ1<θ2<θ p Similarly, θ p The right area is monotonically decreasing, that is, ρ(θ3)>ρ(θ4), θ p <θ3<θ4; the constraint is expressed as follows:

[0033] Gρ≤0

[0034] in, SC is Ω C The number of sampling angles in ;

[0035] Step 1-3, introduce the upper bound of array amplitude and phase errors and construct array error constraints;

[0036] Introducing array amplitude and phase errors, expressed as Where N is the total number of array elements, g en (θ) is the amplitude additive error of the nth (n=1,2,…,N) array element at θ, is the additive phase error of the nth array element at θ;

[0037] Set the ideal array steering vector to The real array steering vector is expressed as in is the ideal array steering vector, x n (n=1,2,...,N) is the array element position, λ is the wavelength, θ is the pitch angle of the incoming wave; the construction error constraint condition is ||a e (θ)||2≤ε; where ||||2 represents the second norm and ε is the upper bound of the second norm of the error vector;

[0038] Step 1-4, construct the main lobe convex constraint condition of the array pattern;

[0039] After introducing the array amplitude and phase errors, the main lobe amplitude constraint is The main lobe amplitude constraint is non-convex, so it is transformed to give it a convex property; it is known that at the main lobe pointing angle, the main lobe constraint term is not sensitive to an additional phase factor on w; therefore, by adjusting the phase factor, Converted to real number form; so the main lobe amplitude pattern is rewritten as Re(w H a(θ)), where Re() is the real part operation; therefore, the error robust amplitude constraint in the main lobe region is rewritten as a convex constraint form

[0040] Steps 1-5, constructing convex constraints for the pattern synthesis optimization problem in the presence of array errors;

[0041] Finally, the expressions for constructing the constraints for different angles of the directional pattern are as follows:

[0042]

[0043] ρ(θ)≥0∈Ω C

[0044] Gρ≤0

[0045] The specific implementation process of step 2 is further given:

[0046] The problem of minimizing the logarithm of the sum of the e-exponentials is used as the cost function to construct a convex optimization problem for the directional pattern synthesis. It is easy to see that the more zero elements there are in the slack variable ρ, the more angles are included in the sidelobe area, that is, the narrower the main lobe. Therefore, the cost function can be set to the l0 norm of ρ; the minimization of the l0 norm is a non-convex problem, which is converted into a convex problem of minimizing the logarithm of the sum of the e-exponentials; therefore, the minimization of the following function is constructed The convex optimization problem with array amplitude and phase errors is expressed as:

[0047]

[0048] ρ(θ)≥0θ∈Ω C

[0049] Gρ≤0

[0050] The present invention is described in detail below with reference to specific embodiments.

[0051] Example

[0052] The present invention proposes a high-gain, low-sidelobe, robust spot beam synthesis method based on sparse optimization, which has the characteristics of being robust to pattern distortion under array error conditions. Figure 1 This example uses an N = 20 array element spacing of half a wavelength uniform linear array, the unit antenna is an isotropic omnidirectional antenna, and the mutual coupling between array elements is not considered. This example achieves a beam pointing of θp = 20°, and a sidelobe angle area of ​​Ω SThe narrowest beam synthesis is [-90°, 5°] ∪ [35°, 90°], and the power level in the sidelobe area is required to be lower than -20dB. The upper limit of the error amplitude is set to ε = 0.2. The number of sampling points of the directional pattern is selected to be 181 points, that is, 1° is a step. The simulation results are obtained using 100 Monte Carlo experiments.

[0053] The implementation of the high-gain, low-sidelobe, robust spot beam synthesis method based on sparse optimization in this uniform linear array with N=20 elements includes the following steps:

[0054] Step 1: Point the beam according to θ p = 20° and the expected sidelobe level distribution, the expected amplitude pattern is determined as Will Introduce it into the desired pattern, and according to the side lobe angle area size and pointing angle θ p = 20°, the ideal spot beam width pointing to this angle is 5.28°. Based on approximately 6 times the ideal spot beam width, the number of sampling points S in the area near the pointing angle can be taken as C =29, thus the monotone constraint matrix of the slack variables can be established as

[0055]

[0056] Its dimensions are 28×29.

[0057] Step 2: Introduce array amplitude and phase error a e (θ), constraint ||a e (θ)||2≤0.2. The corresponding directional vector error of a single antenna unit is an amplitude error with a maximum amplitude of 0.04 at each angle and a phase error of 11.53°. From this, the constraint conditions of the convex optimization problem of the directional pattern synthesis under the existence of amplitude and phase errors are:

[0058] Re(w H a(θ))≥F d (θ)+ε||w||2θ=20°

[0059] |w H a(θ)|≤F d (θ)-ε||w||2θ∈[-90°,5°]∪[35°,90°]

[0060] |w H a(θ)|≤F d (θ)+ρ(θ)-ε||w||2θ∈[6°,19°]∪[21°,34°]

[0061] ρ(θ)≥0θ∈[6°,34°]

[0062] Gρ≤0

[0063] in, Ideal directional vector

[0064] Step 3: Take the logarithm of the sum of the e exponentials of the slack variables, that is As the minimization optimization target, a convex optimization problem of the pattern synthesis with the narrowest main lobe area as the goal is constructed:

[0065]

[0066] ρ(θ)≥0θ∈[6°,34°]

[0067] Gρ≤0

[0068] Step 4: Use the CVX toolbox in MATLAB to solve.

[0069] For this example, Figure 2 The optimized radiation pattern of the method of the present invention in the presence of amplitude and phase errors is given. It can be seen that in 100 Monte Carlo experiments, the radiation pattern can always ensure that the sidelobe level is lower than -22dB, meeting the sidelobe power constraint of -20dB, and the main lobe width is 6.4°, which shows that the method of the present invention can effectively suppress the influence of the array amplitude and phase errors in the sidelobe area; Figure 3-1 and Figure 3-2 The comparison between the directional patterns optimized by the existing synthesis method and the directional patterns of the present invention is given respectively when there is no error and when there is an error and the sidelobe performance requirements are similar. Among them, 3-1(a) and 3-2(a) are the comprehensive directional patterns of the method of the present invention, and 3-1(b) and 3-2(b) are the beam synthesis directional patterns of the existing directional pattern synthesis method when the sidelobe constraints are set to -26dB and -30dB, respectively. It can be seen that when the sidelobe constraint is set to -26dB, the simulation directional patterns of the existing directional pattern synthesis method and the method of the present invention are basically the same when there is no amplitude and phase error. However, when there is an amplitude and phase error, the sidelobe level of the existing directional pattern synthesis method is higher, which is about 2dB higher than the sidelobe of the latter, and there are some angles higher than -20dB. When the sidelobe constraint is set to -30dB, the directional pattern sidelobe level of the existing directional pattern synthesis method is slightly worse than that of the method of the present invention, and the main lobe of the former is wider by about 0.3° than the latter, which means that the beam gain of the existing directional pattern synthesis algorithm is lower than that of the method of the present invention. According to the above simulation, it can be seen that the method of the present invention is superior to the existing pattern synthesis algorithm in two aspects: the robustness of pattern control under the condition of array element amplitude and phase errors, and the control of the narrowest main lobe pattern.

[0070] The above-described embodiments of the present application do not constitute a limitation on the protection scope of the present application.

Claims

1. Array error robust narrowest beam synthesis method based on sparse optimization, characterized by: The method comprises: Step 1, determine the pattern power constraint according to the beam pointing and the expected sidelobe level distribution, introduce slack variables and construct convex constraint conditions for pattern synthesis; Step 2, construct a cost function based on the logarithm minimization problem of the sum of the e-exponentials of the slack variables, and build a convex optimization problem for directional pattern synthesis; Step 3: Use the convex optimization solving tool to solve the convex optimization problem of pattern synthesis.

2. The method according to claim 1, characterized in that Step 1: Determine the pattern power constraint based on the beam pointing and the expected sidelobe level distribution, introduce slack variables and construct convex constraints for pattern synthesis, including: Step 1-1, determine the desired pattern F according to the beam pointing to the sidelobe level distribution d (θ); Step 1-2, introduce the slack variable ρ(θ) and modify the desired pattern to F d2 (θ), construct the monotonicity constraint of the directional graph; Step 1-3, introduce the upper bound of array amplitude and phase errors and construct array error constraints; Step 1-4, construct the main lobe convex constraint condition of the array pattern; Steps 1-5, construct convex constraints for the pattern synthesis optimization problem in the presence of array errors.

3. The method according to claim 2, characterized in that Step 1-1, determine the desired pattern F according to the beam pointing to the sidelobe level distribution d (θ); including: Assume that the desired beam direction of the pattern is θ p , where θ p is the angle predetermined by the project indicator; the ideal expectation of spot beamforming is that the directional pattern in other angle areas except the pointing angle must meet the low sidelobe power constraint; therefore, the ideal expected directional pattern is expressed as Among them, M u is the expected amplitude of the main lobe, S u is the expected amplitude of the side lobe, Ω all is the angle area of ​​the entire integrated pattern, where \ represents except, that is, except Ω all Except for θ p Outside angle area.

4. The method according to claim 3, characterized in that Step 1-2, introduce the slack variable ρ(θ) and modify the desired pattern to F d2 (θ), construct the monotonicity constraint of the directional graph, including: At the pointing angle θ p The slack variable ρ(θ) is introduced in the nearby area to relax the power constraint of the pattern in the corresponding angle area; and when ρ(θ) = 0, θ is included in the sidelobe area; the modified desired pattern is expressed as: Among them, Ω c is the angular area near the beam pointing, Ω s is the side lobe angle area; Next, we use the monotonicity of the main lobe area of ​​the pattern to impose constraints on ρ(θ); since θ p The directional pattern level in the left angle area is monotonically increasing, that is, ρ(θ1)<ρ(θ2), where θ1<θ2<θ p Similarly, θ p The right area is monotonically decreasing, that is, ρ(θ3)>ρ(θ4), θ p <θ3<θ4; the constraint is expressed as follows: Gρ≤0 in, SC is Ω C The number of sampling angles in .

5. The method according to claim 4, characterized in that Step 1-3, introduce the upper bound of array amplitude and phase errors and construct array error constraints; including: Introducing array amplitude and phase errors, expressed as Where N is the total number of array elements, g en (θ) is the amplitude additive error of the nth (n=1, 2, ..., N) array element at θ, is the additive phase error of the nth array element at θ; Set the ideal array steering vector to The real array steering vector is expressed as in is the ideal array steering vector, x n (n=1, 2, ..., N) is the array element position, λ is the wavelength, and θ is the pitch angle of the incoming wave; the construction error constraint condition is ||a e (θ)||2≤ε; where || ||2 represents the second norm, and ε is the upper bound of the second norm of the error vector. Step 1-4, construct the main lobe convex constraint condition of the array pattern; After introducing the array amplitude and phase errors, the main lobe amplitude constraint is The main lobe amplitude constraint is non-convex, so it is transformed to give it a convex property; it is known that at the main lobe pointing angle, the main lobe constraint term is not sensitive to an additional phase factor on w; therefore, by adjusting the phase factor, Converted to real number form; so the main lobe amplitude pattern is rewritten as Re(w H a(θ)), where Re() is the real part operation; therefore, the error robust amplitude constraint in the main lobe region is rewritten as a convex constraint form 6. The method according to claim 5, characterized in that Steps 1-5, constructing convex constraints for the pattern synthesis optimization problem in the presence of array errors, including: Finally, the expressions for constructing the constraints for different angles of the directional pattern are as follows:

7. The method according to claim 6, characterized in that Step 2, constructing a cost function based on the logarithm minimization problem of the sum of the e-exponentials of the slack variables, and building a convex optimization problem for directional pattern synthesis; including: The problem of minimizing the logarithm of the sum of the e-exponentials is used as the cost function to construct a convex optimization problem for directional pattern synthesis; therefore, the cost function can be set to the 10-norm of ρ; the minimization of the 10-norm is a non-convex problem, which is transformed into a convex problem of minimizing the logarithm of the sum of the e-exponentials; therefore, the minimization of the following function is constructed The convex optimization problem with array amplitude and phase errors is expressed as: ρ(θ)≥0θ∈Ω C Gρ≤0.

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