CN112162266A - Conformal array two-dimensional beam optimization method based on convex optimization theory - Google Patents

Conformal array two-dimensional beam optimization method based on convex optimization theory Download PDF

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CN112162266A
CN112162266A CN202011038156.0A CN202011038156A CN112162266A CN 112162266 A CN112162266 A CN 112162266A CN 202011038156 A CN202011038156 A CN 202011038156A CN 112162266 A CN112162266 A CN 112162266A
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array
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main lobe
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CN112162266B (en
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梁浩前
王晓庆
陈卫东
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CETC 54 Research Institute
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/52—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S15/00
    • G01S7/539—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S15/00 using analysis of echo signal for target characterisation; Target signature; Target cross-section
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S15/00—Systems using the reflection or reradiation of acoustic waves, e.g. sonar systems
    • G01S15/006—Theoretical aspects

Abstract

The invention discloses a conformal array two-dimensional wave beam optimization method based on a convex optimization theory, and belongs to the technical field of underwater acoustic array signal processing. The method comprises the following steps: dividing a main lobe area and a side lobe area of the conformal array scanning space; initializing a half main lobe width, and setting the narrowest main lobe width under an expected side lobe level through iterative solution; and expressing the beam forming optimization problem into a convex optimization second-order cone constraint form, and solving the optimal solution of the weighting vector by using an inner point method so as to obtain a uniform low-sidelobe two-dimensional beam response diagram. The two-dimensional beam pattern obtained by the invention has lower side lobe level, the side lobe is uniform side lobe, and the two-dimensional beam pattern has more excellent beam directivity. The constraint equation designed on the basis of the MVDR method enables the two-dimensional beam forming to obtain higher array gain compared with the traditional method. In addition, the constraint on the weighting vector norm in the invention enables the two-dimensional beam forming to have higher robustness compared with the traditional method.

Description

Conformal array two-dimensional beam optimization method based on convex optimization theory
Technical Field
The invention belongs to the technical field of underwater acoustic array signal processing, and particularly relates to a conformal array two-dimensional beam optimization method based on a convex optimization theory.
Background
When the active sonar is used for detecting water surface targets such as ships, the water surface targets are not at the same height as the sonar platform, and the sonar needs to search in the horizontal direction and in the pitching direction. The conformal array sonar can realize space all-round scanning, and the purpose of reducing algorithm operand and sonar equipment complexity is worked out in the traditional processing means, is equivalent to the one-dimensional array with conformal array and carries out one-dimensional beam forming, is about to each array element series connection processing of every pitch direction is equivalent to an array element, and every equivalent array element has certain directive property in the pitch direction. However, the pitch direction detection range of such processing is limited to about 0 °, the anti-pitch direction interference performance is poor, the beam side lobe level is high, and the performance is deteriorated when the acoustic propagation path is lost.
A two-dimensional beam forming method based on a product theorem decomposes an area array into a combination of a linear array and a linear array, and respectively carries out beam forming on the two linear arrays to finally obtain a beam pattern of a three-dimensional array, but the method is not suitable for conformal arrays with randomly distributed array element spaces. Numerical synthesis methods apply several virtual interferences in the side lobe region based on minimum variance distortion free response (MVDR) beamforming. These interferences may reduce the beam response in the corresponding direction and increasing the interference strength may reduce the corresponding beam response. And adjusting the interference intensity through an iterative method, and controlling the beam side lobe to reach an expected side lobe level. The numerical synthesis method and the side lobe control method based on the self-adaptive beam forming have the problems that iteration step length is difficult to select, strict control on side lobes is difficult to guarantee, main lobes are easy to widen obviously due to the fact that the iteration process is free of constraint on the main lobes, the narrowest main lobe width under a given side lobe level cannot be obtained, the problems are more obvious when the numerical synthesis method is applied to two-dimensional beam forming, and the algorithm cannot be converged frequently. Furthermore, such methods lack considerations for array gain and robustness in controlling the sidelobes.
With the development of optimization theory and the progress of computer technology, a method for performing beam optimization and sidelobe control by using convex optimization theory appears. The convex optimization method can obtain the global optimal solution of the problem, and can avoid the problems of searching step length selection, trapping in local extremum and the like. The convex optimization problem can be conveniently solved according to the existing interior point method. Therefore, the convex optimization theory and the solving algorithm thereof provide a new solving idea and direction for the two-dimensional beam optimization and the sidelobe control of the conformal array.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a conformal array two-dimensional beam optimization method based on a convex optimization theory, which solves the problems that when the conformal array performs two-dimensional beam forming, the side lobe level is too high and the beam stability is difficult to control, and simultaneously improves the array gain and the robustness of beam forming.
The technical problem to be solved by the invention is realized by the following technical scheme:
a conformal array two-dimensional wave beam optimization method based on a convex optimization theory comprises the following steps:
(1) performing array flow pattern modeling in a conformal array scanning space to obtain a two-dimensional array flow pattern matrix A;
(2) obtaining a beam response of a target direction according to the azimuth angle and the pitch angle pointed by the target;
(3) dividing a main lobe area and a side lobe area of the conformal array scanning space;
(4) initializing the widths of the azimuth half main lobe and the pitching half main lobe, and setting an expected side lobe level;
(5) calculating the narrowest half main lobe width with a solution through loop iteration, and obtaining a main lobe region and a side lobe region under the narrowest half main lobe width;
(6) solving convex optimization second-order cone constraint by using an inner point method to obtain a beam forming weighting vector w;
(7) and calculating the beam response formed by the two-dimensional beam according to the weighting vector w and the two-dimensional array flow pattern matrix A.
Further, the specific mode of the step (1) is as follows:
(101) uniformly discretizing the radiation space into theta according to azimuth angle and pitch angle1,θ2,…,θP、
Figure BDA0002705780640000021
Wherein theta is1,θ2,…,θP∈[-90°,90°],
Figure BDA0002705780640000022
Subscripts P and Q are the number of discrete azimuth angles and pitch angles, respectively;
(102) for an N-element conformal array, the three-dimensional coordinates of hydrophone array element i are represented as ri＝[xi,yi,zi]TI is more than or equal to 1 and less than or equal to N; for the direction of radiation space
Figure BDA0002705780640000023
The unit direction vector is expressed as
Figure BDA0002705780640000024
T is the transposition of the matrix;
(103) taking the origin of coordinates as a reference point, the time delay of the array element i relative to the origin of coordinates is:
Figure BDA0002705780640000025
the N-dimensional steering vector of the array is:
Figure BDA0002705780640000026
wherein c is underwater sound velocity, f is signal center frequency, j is imaginary number unit, G1、G2,...,GNDirectivity of conformal array hydrophone
Figure BDA0002705780640000031
Phi is more than 180 degrees and less than or equal to 180 degrees, and phi is an included angle between the array element sound receiving main shaft and the signal incidence direction;
thereby, obtaining an azimuth and elevation two-dimensional array flow pattern matrix of the matrix:
Figure BDA0002705780640000032
and obtaining the array pairs
Figure BDA0002705780640000033
The beam response in the direction is:
Figure BDA0002705780640000034
where p (-) is the beam response function and the superscript H denotes the conjugate transpose.
Further, in step (3), the main lobe region θmain,
Figure BDA0002705780640000035
Comprises the following steps:
θmain＝(θs-αa,θs+αa)，
Figure BDA0002705780640000036
side lobe region thetaside,
Figure BDA0002705780640000037
Comprises the following steps:
θside＝[(-90°,θs-αa)∪(θs+αa,90°)]，
Figure BDA0002705780640000038
and
θside＝(θs-αa,θs+αa)，
Figure BDA0002705780640000039
wherein the content of the first and second substances,
Figure BDA00027057806400000310
is the target direction, αaIn azimuth half main lobe width, αeIs the pitch half main lobe width.
Further, the specific mode of the step (5) is as follows:
(501) for half mainlobe widths, the constraint equation is solved by the interior point method:
Figure BDA00027057806400000311
Figure BDA00027057806400000312
θp∈θside,
Figure BDA00027057806400000313
a subscript P1,.. q.
Wherein the content of the first and second substances,
Figure BDA00027057806400000314
response of the beam in the target direction, xipqA desired side lobe level; if the equation has a solution, α ═ α -1, and if the equation has no solution, α ═ α + 1;
(502) iterating the loop of the step (501) until alpha is solved and alpha-1 is not solved, and obtaining the main lobe region theta 'at the moment'main,
Figure BDA00027057806400000315
Comprises the following steps:
θ′main＝(θs-αamin,θs+αamin)，
Figure BDA00027057806400000316
side lobe region theta'side,
Figure BDA00027057806400000317
Comprises the following steps:
θ′side＝[(-90°,θs-αamin)∪(θs+αamin,90°)]，
Figure BDA0002705780640000041
and
θ′side＝(θs-αamin,θs+αamin)，
Figure BDA0002705780640000042
wherein alpha isaminAnd alphaeminThe narrowest azimuth half main lobe width and the narrowest pitching half main lobe width are obtained respectively.
Further, the convex optimization second order cone constraint in step (6) is expressed as:
Figure BDA0002705780640000043
s.t
Figure BDA0002705780640000044
Figure BDA0002705780640000045
θp∈θ′side,
Figure BDA0002705780640000046
a subscript P1,.. q.
Wherein w ═ w1,w2,...,wN]TRepresenting a weight vector, wiIs a weight value of array element i, RnIs a noise covariance matrix;
r in the case of spatial white noisenAs an identity matrix, the second order cone constraint is simplified as:
Figure BDA0002705780640000047
s.t
Figure BDA0002705780640000048
Figure BDA0002705780640000049
θp∈θ′side,
Figure BDA00027057806400000410
a subscript P1,.. q.
And solving the second-order cone constraint through an interior point method to obtain the optimal solution of the weighting vector w under the second-order cone constraint.
Further, the beam response of the two-dimensional beam forming in step (7)
Figure BDA00027057806400000411
The calculation method is as follows:
Figure BDA00027057806400000412
compared with the prior art, the invention has the following advantages:
1. compared with the traditional method, the two-dimensional beam pattern obtained by the invention has lower side lobe level, and the side lobe is uniform side lobe, so that the invention has better beam directivity.
2. The invention solves the global optimal solution through a convex optimization method, and can obtain the narrowest main lobe under the set expected side lobe level.
3. Compared with the conventional method, the second-order cone constraint beam forming method has higher array gain under the same side lobe level.
4. By constraining the norm of the weighting vector, the robustness of beam forming is higher than that of the traditional method.
Drawings
FIG. 1 is a diagram of an array received signal model in an embodiment of the invention;
fig. 2 is a flow chart of two-dimensional beamforming in an embodiment of the invention;
FIG. 3 is a diagram of a division of a main lobe region and a side lobe region in an embodiment of the present invention;
FIG. 4 is a diagram of a conformal array model used in simulation according to an embodiment of the present invention;
FIGS. 5-10 are two-dimensional beamforming beampatterns using Conventional Beamforming (CBF), numerical synthesis (OLEN), and the method of the present invention;
FIGS. 11-14 are graphs comparing the beam optimization results of the method of the present invention with the processing results of the conventional beamforming and numerical synthesis method;
fig. 15 is a comparison of the array gain of the method of the present invention with conventional beamforming and numerically synthesized array gain.
Detailed Description
The technical solution and effects of the present invention will be further described in detail with reference to the accompanying drawings.
A conformal array two-dimensional wave beam optimization method based on a convex optimization theory is provided, and increases the constraint of a side lobe of wave beam response formed by a two-dimensional wave beam on the basis of high array gain obtained by a minimum variance distortion free response (MVDR) wave beam forming method, and represents the constraint as a convex optimization second-order cone constraint problem. The method specifically comprises the following steps:
(1) performing array flow pattern modeling in a conformal array scanning space to obtain a two-dimensional array flow pattern matrix A;
(2) obtaining a beam response of a target direction according to the azimuth angle and the pitch angle pointed by the target;
(3) dividing a main lobe area and a side lobe area of the conformal array scanning space;
(4) initializing the widths of the azimuth half main lobe and the pitching half main lobe, and setting an expected side lobe level;
(5) calculating the narrowest half main lobe width with a solution through loop iteration, and obtaining a main lobe region and a side lobe region under the narrowest half main lobe width;
(6) solving convex optimization second-order cone constraint by using an interior point method to obtain a weighting vector w;
(7) and calculating the beam response formed by the two-dimensional beam according to the weighting vector w and the two-dimensional array flow pattern matrix A.
Wherein, the step (1) comprises the following steps:
(101) uniformly discretizing the radiation space into theta according to azimuth angle and pitch angle1,θ2,…,θP、
Figure BDA0002705780640000051
Wherein theta is1,θ2,…,θP∈[-90°,90°],
Figure BDA0002705780640000052
P and Q are the number of discrete azimuth angles and pitch angles respectively;
(102) for an N-element conformal array, the three-dimensional coordinates of hydrophone array element i are represented as ri＝[xi,yi,zi]TI is more than or equal to 1 and less than or equal to N; for the direction of radiation space
Figure BDA0002705780640000061
The unit direction vector is expressed as
Figure BDA0002705780640000062
T is the transposition of the matrix;
(103) taking the origin of coordinates as a reference point, the time delay of the array element i relative to the origin of coordinates is:
Figure BDA0002705780640000063
the N-dimensional steering vector of the array is:
Figure BDA0002705780640000064
wherein c is underwater sound velocity, f is signal center frequency, j is imaginary number unit, G1、G2,...,GNDirectivity of conformal array hydrophone
Figure BDA0002705780640000065
Phi is more than 180 degrees and less than or equal to 180 degrees, and phi is an included angle between the array element sound receiving main shaft and the signal incidence direction;
thereby, obtaining an azimuth and elevation two-dimensional array flow pattern matrix of the matrix:
Figure BDA0002705780640000066
and obtaining the array pairs
Figure BDA0002705780640000067
The beam response in the direction is:
Figure BDA0002705780640000068
wherein the superscript H denotes the conjugate transpose.
In step (3), the main lobe region θmain,
Figure BDA0002705780640000069
Comprises the following steps:
θmain＝(θs-αa,θs+αa)，
Figure BDA00027057806400000610
side lobe region thetaside,
Figure BDA00027057806400000611
Comprises the following steps:
θside＝[(-90°,θs-αa)∪(θs+αa,90°)]，
Figure BDA00027057806400000612
and
θside＝(θs-αa,θs+αa)，
Figure BDA00027057806400000613
wherein the content of the first and second substances,
Figure BDA0002705780640000071
is the target direction, αaIn azimuth half main lobe width, αeIs the pitch half main lobe width.
The iteration method in the step (5) specifically comprises the following steps:
(501) for half mainlobe widths, the constraint equation is solved by the interior point method:
Figure BDA0002705780640000073
θp∈θside,
Figure BDA0002705780640000074
p＝1,...,P,q＝1,...,Q
wherein the content of the first and second substances,
Figure BDA0002705780640000075
response of the beam in the target direction, xipqA desired side lobe level; if the equation has a solution, α ═ α -1, and if the equation has no solution, α ═ α + 1;
(502) iterating the loop of the step (501) until alpha is solved and alpha-1 is not solved, and obtaining the main lobe region theta 'at the moment'main,
Figure BDA0002705780640000076
Comprises the following steps:
θ′main＝(θs-αamin,θs+αamin)，
Figure BDA0002705780640000077
side lobe region thetas′ide,
Figure BDA0002705780640000078
Comprises the following steps:
θ′side＝[(-90°,θs-αamin)∪(θs+αamin,90°)]，
Figure BDA0002705780640000079
and
θ′side＝(θs-αamin,θs+αamin)，
Figure BDA00027057806400000710
wherein alpha isaminAnd alphaeminThe narrowest azimuth half main lobe width and the narrowest pitching half main lobe width are obtained respectively.
The convex optimization second-order cone constraint in the step (6) is expressed as:
Figure BDA00027057806400000711
s.t
Figure BDA00027057806400000712
Figure BDA00027057806400000713
θp∈θ′side,
Figure BDA00027057806400000714
p＝1,...,P,q＝1,...,Q
wherein w ═ w1,w2,...,wN]TRepresenting a weighted vector, wiIs a weight value of array element i, RnIs a noise covariance matrix;
r in the case of spatial white noisenAs an identity matrix, the second order cone constraint is simplified as:
Figure BDA00027057806400000715
s.t
Figure BDA00027057806400000716
Figure BDA00027057806400000717
θp∈θ′side,
Figure BDA00027057806400000718
p＝1,...,P,q＝1,...,Q
and solving the second-order cone constraint through an interior point method to obtain the optimal solution of the weighting vector w under the second-order cone constraint.
The calculation formula of the beam response of the two-dimensional beam forming in the step (7) is as follows:
Figure BDA0002705780640000081
the method can be used for the optimization design of the transmitting and receiving beam patterns of the active conformal array sonar system, and the beam directivity of the sonar system is improved.
The following is a more specific example:
FIG. 1 is a diagram of an array received signal model. The figure shows a model of a conformal array receiving signal by taking an array of N-element space conformal arrays as an example. The array receives signals from far field points P. Theta and
Figure BDA0002705780640000082
respectively representing the incidence azimuth angle in the horizontal direction and the incidence pitch angle in the depth direction of the incoming wave signal. Assuming that the conformal array radiates only half-space forward, θ e [ -90 °,90 °]、
Figure BDA0002705780640000083
The specific flow of the conformal array two-dimensional beam forming method in this example is shown in fig. 2, and the implementation steps are as follows:
step 1: referring to FIG. 1, a conformal array flow pattern is modeled.
(101) Uniformly discretizing the radiation space into theta according to azimuth angle and pitch angle1,θ2,…,θP、
Figure BDA0002705780640000084
Wherein theta is1,θ2,…,θP∈[-90°,90°],
Figure BDA0002705780640000085
P and Q are the number of discrete azimuth angles and pitch angles respectively;
(102) for an N-element conformal array, the three-dimensional coordinates of hydrophone array element i are represented as ri＝[xi,yi,zi]TI is more than or equal to 1 and less than or equal to N; for the direction of radiation space
Figure BDA0002705780640000086
The unit direction vector is expressed as
Figure BDA0002705780640000087
T is the transposition of the matrix;
(103) taking the origin of coordinates as a reference point, the time delay of the array element i relative to the origin of coordinates is:
Figure BDA0002705780640000088
the N-dimensional steering vector of the array is:
Figure BDA0002705780640000089
wherein c is underwater soundSpeed, f is the signal center frequency, j is the imaginary unit, G1、G2,...,GNDirectivity of conformal array hydrophone
Figure BDA0002705780640000091
Phi is more than 180 degrees and less than or equal to 180 degrees, and phi is an included angle between the array element sound receiving main shaft and the signal incidence direction;
thereby, obtaining an azimuth and elevation two-dimensional array flow pattern matrix of the matrix:
Figure BDA0002705780640000092
and obtaining the array pairs
Figure BDA0002705780640000093
The beam response in the direction is:
Figure BDA0002705780640000094
wherein the superscript H denotes the conjugate transpose.
Step 2: obtaining beam response in the target (incoming wave) direction
Figure BDA0002705780640000095
And step 3: referring to fig. 3, the conformal array scanning space is divided into a main lobe area and a side lobe area.
(301) Set the target direction as
Figure BDA0002705780640000096
Width of azimuth half main lobe is alphaaThe width of half main lobe of pitch is alphaeMain lobe region θmain,
Figure BDA0002705780640000097
Comprises the following steps:
θmain＝(θs-αa,θs+αa)，
Figure BDA0002705780640000098
side lobe region thetaside,
Figure BDA0002705780640000099
Comprises the following steps:
θside＝[(-90°,θs-αa)∪(θs+αa,90°)]，
Figure BDA00027057806400000910
and
θside＝(θs-αa,θs+αa)，
Figure BDA00027057806400000911
and 4, step 4: the half main lobe width sum of azimuth and pitch is initialized, setting the desired side lobe level.
And 5: the iterative solution is performed according to the following manner:
(501) for half mainlobe widths, the constraint equation is solved by the interior point method:
Figure BDA00027057806400000912
Figure BDA00027057806400000913
θp∈θside,
Figure BDA00027057806400000914
p＝1,...,P,q＝1,...,Q
wherein ξpqA desired side lobe level; if the equation has a solution, α ═ α -1, and if the equation has no solution, α ═ α + 1;
(502) iterating the loop of the step (501) until alpha is solved and alpha-1 is not solved, and obtaining the main lobe region theta 'at the moment'main,
Figure BDA00027057806400000915
Comprises the following steps:
θ′main＝(θs-αamin,θs+αamin)，
Figure BDA00027057806400000916
side lobe region theta'side,
Figure BDA0002705780640000101
Comprises the following steps:
θs′ide＝[(-90°,θs-αamin)∪(θs+αamin,90°)]，
Figure BDA0002705780640000102
and
θ′side＝(θs-αamin,θs+αamin)，
Figure BDA0002705780640000103
wherein alpha isaminAnd alphaeminThe narrowest azimuth half main lobe width and the narrowest pitching half main lobe width are obtained respectively.
Step 6: and (5) solving convex optimization second-order cone constraint to obtain a beam forming weighting vector w.
The beam forming method based on MVDR has higher array gain, and side lobe control is applied to the beam forming method:
Figure BDA0002705780640000104
θp∈θ′side,
Figure BDA0002705780640000105
p＝1,...,P,q＝1,...,Q
forming the MVDR wave beam applying the side lobe control into a second-order cone constraint form of a convex optimization problem:
Figure BDA0002705780640000106
s.t
Figure BDA0002705780640000107
Figure BDA0002705780640000108
θp∈θ′side,
Figure BDA0002705780640000109
p＝1,...,P,q＝1,...,Q
wherein w ═ w1,w2,...,wN]TRepresenting a weighted vector, wiIs a weight value of array element i, RnIs a noise covariance matrix. R in the case of spatial white noisenAs an identity matrix, the second order cone constraint is simplified as:
Figure BDA00027057806400001010
s.t
Figure BDA00027057806400001011
Figure BDA00027057806400001012
θp∈θ′side,
Figure BDA00027057806400001013
p＝1,...,P,q＝1,...,Q
weighted vector norm | | w | | non-woven phosphor2The lower, the more robust the beamforming. By adding the constraint on the norm of the weighting vector | | w | | luminance2Gamma is less than or equal to gamma, and gamma is a set scalar quantity, can be increasedRobustness of beamforming. Objective function of above second order cone constraint
Figure BDA00027057806400001014
Non-woven cells with constraint | | w |)2The ≦ γ action repetition, i.e., this second order cone constraint, may already improve the robustness of beamforming.
The second-order cone constraint is solved through an interior point method, and the optimal solution of the weighting vector w under the second-order cone constraint can be obtained.
And 7: and according to the solved optimal weighting vector w and the two-dimensional array flow pattern matrix A, calculating to obtain two-dimensional beam forming beam response by the following formula:
Figure BDA0002705780640000111
the effect of the method can be illustrated by the following simulation:
1. simulation condition and method
An 8-row 216-membered ring belt conformal array was set up as shown in fig. 4. The array surface is hemispherical, 8 rows of curve arrays are arranged on the hemispherical surface in parallel to the horizontal plane, each row of 27 array elements are arranged, the distance between each array element and the origin is 220mm, and the distance between the array elements is 25 mm. At a signal frequency of 30kHz and an acoustic velocity of 1500m/s in water, the half wavelength lambda/2 of the signal is 25 mm. The array element interval is half wavelength, which accords with the sampling theorem. The target orientations were [0 °,0 ° ], [60 °,0 ° ].
2. Simulation content and results
On the premise of not changing the simulation conditions, the conformal array two-dimensional beam optimization method based on the convex optimization theory provided herein is compared with the Conventional Beam Forming (CBF) and numerical synthesis (OLEN) methods, and the obtained two-dimensional beam patterns are respectively shown in fig. 5 to 10. The results shown in fig. 11 to 14 can be obtained by performing azimuth beam slicing and elevation beam slicing in the target pointing direction on each two-dimensional beam pattern, and comparing the one-dimensional beam patterns of each algorithm. The two-dimensional beam pattern obtained by the method has lower side lobe level, the side lobe is uniform side lobe, and the beam directivity is more excellent. During simulation, the expected side lobe level is set to be-30 dB, and the method can be found out that the expected side lobe level can be set and a beam pattern meeting the setting requirement can be obtained.
On the premise of not changing the beam direction, the high array gain characteristic of the method of the invention at different frequencies is continuously examined. The results shown in fig. 10 were obtained by simulating the difference between the input and output signal-to-noise ratios of the three methods, i.e., the array gain, at different frequencies. At different frequencies, the method has higher array gain than the other two methods.
In summary, the method of the present invention is a side lobe control high gain high robustness two-dimensional beam optimization method suitable for any sensor array, and the method mainly includes the following steps: dividing a main lobe area and a side lobe area of the conformal array scanning space; initializing a half main lobe width, and setting the narrowest main lobe width under an expected side lobe level through iterative solution; and expressing the beam forming optimization problem into a convex optimization second-order cone constraint form, and solving the optimal solution of the weighting vector by using an inner point method so as to obtain a uniform low-sidelobe two-dimensional beam response diagram.
The two-dimensional beam pattern obtained by the method has lower side lobe level, the side lobe is uniform side lobe, and the beam directivity is more excellent. In addition, the method of the invention can set the expected side lobe level and can obtain the beam pattern meeting the setting requirement.
The constraint equation designed on the basis of the MVDR method enables the beam forming to obtain higher array gain compared with the traditional method. The constraint of the weighting vector norm of the invention ensures that the beam forming has higher robustness compared with the traditional method, and effectively solves the problems of high sidelobe and poor beam directivity of the two-dimensional beam pattern transmitted and received by the active conformal array sonar system.

Claims (6)

1. A conformal array two-dimensional wave beam optimization method based on a convex optimization theory is characterized by comprising the following steps:
(1) performing array flow pattern modeling in a conformal array scanning space to obtain a two-dimensional array flow pattern matrix A;
(2) obtaining a beam response of a target direction according to the azimuth angle and the pitch angle pointed by the target;
(3) dividing a main lobe area and a side lobe area of the conformal array scanning space;
(4) initializing the widths of the azimuth half main lobe and the pitching half main lobe, and setting an expected side lobe level;
(5) calculating the narrowest half main lobe width with a solution through loop iteration, and obtaining a main lobe region and a side lobe region under the narrowest half main lobe width;
(6) solving convex optimization second-order cone constraint by using an inner point method to obtain a beam forming weighting vector w;
(7) and calculating the beam response formed by the two-dimensional beam according to the weighting vector w and the two-dimensional array flow pattern matrix A.
2. The conformal array two-dimensional beam optimization method based on the convex optimization theory according to claim 1, wherein the specific manner of the step (1) is as follows:
(101) uniformly discretizing the radiation space into theta according to azimuth angle and pitch angle1,θ2,…,θP、
Figure FDA0002705780630000011
Wherein theta is1,θ2,…,θP∈[-90°,90°],
Figure FDA0002705780630000012
Subscripts P and Q are the number of discrete azimuth angles and pitch angles, respectively;
(102) for an N-element conformal array, the three-dimensional coordinates of hydrophone array element i are represented as ri＝[xi,yi,zi]TI is more than or equal to 1 and less than or equal to N; for the direction of radiation space
Figure FDA0002705780630000013
The unit direction vector is expressed as
Figure FDA0002705780630000014
T is the transposition of the matrix;
(103) taking the origin of coordinates as a reference point, the time delay of the array element i relative to the origin of coordinates is:
Figure FDA0002705780630000015
the N-dimensional steering vector of the array is:
Figure FDA0002705780630000016
wherein c is underwater sound velocity, f is signal center frequency, j is imaginary number unit, G1、G2,...,GNDirectivity of conformal array hydrophone
Figure FDA0002705780630000021
Phi is an included angle between the array element sound receiving main shaft and the signal incidence direction;
thereby, obtaining an azimuth and elevation two-dimensional array flow pattern matrix of the matrix:
Figure FDA0002705780630000022
and obtaining the array pairs
Figure FDA0002705780630000023
The beam response in the direction is:
Figure FDA0002705780630000024
where p (-) is the beam response function and the superscript H denotes the conjugate transpose.
3. The conformal array two-dimensional beam optimization method based on convex optimization theory as claimed in claim 2, wherein in step (3), the main lobe area θ ismain,
Figure FDA0002705780630000025
Comprises the following steps:
θmain＝(θs-αa,θs+αa)，
Figure FDA0002705780630000026
side lobe region thetaside,
Figure FDA0002705780630000027
Comprises the following steps:
θside＝[(-90°,θs-αa)∪(θs+αa,90°)]，
Figure FDA0002705780630000028
and
θside＝(θs-αa,θs+αa)，
Figure FDA0002705780630000029
wherein the content of the first and second substances,
Figure FDA00027057806300000210
is the target direction, αaIn azimuth half main lobe width, αeIs the pitch half main lobe width.
4. The conformal array two-dimensional beam optimization method based on the convex optimization theory according to claim 3, wherein the specific manner of the step (5) is as follows:
(501) for half mainlobe widths, the constraint equation is solved by the interior point method:
Figure FDA00027057806300000211
Figure FDA00027057806300000212
θp∈θside,
Figure FDA00027057806300000213
a subscript P1,.. q.
Wherein the content of the first and second substances,
Figure FDA00027057806300000214
response of the beam in the target direction, xipqA desired side lobe level; if the equation has a solution, α ═ α -1, and if the equation has no solution, α ═ α + 1;
(502) iterating the loop of the step (501) until alpha is solved and alpha-1 is not solved, and obtaining the main lobe region theta 'at the moment'main,
Figure FDA00027057806300000215
Comprises the following steps:
θ′main＝(θs-αamin,θs+αamin)，
Figure FDA0002705780630000031
side lobe region theta'side,
Figure FDA0002705780630000032
Comprises the following steps:
θ′side＝[(-90°,θs-αamin)∪(θs+αamin,90°)]，
Figure FDA0002705780630000033
and
θ′side＝(θs-αamin,θs+αamin)，
Figure FDA0002705780630000034
wherein alpha isaminAnd alphaeminThe narrowest azimuth half main lobe width and the narrowest pitching half main lobe width are obtained respectively.
5. The conformal array two-dimensional beam optimization method based on convex optimization theory according to claim 4, wherein the convex optimization second order cone constraint in step (6) is expressed as:
Figure FDA0002705780630000035
Figure FDA0002705780630000036
Figure FDA0002705780630000037
θp∈θ′side,
Figure FDA0002705780630000038
a subscript P1,.. q.
Wherein w ═ w1,w2,...,wN]TRepresenting a weight vector, wiIs a weight value of array element i, RnIs a noise covariance matrix;
r in the case of spatial white noisenAs an identity matrix, the second order cone constraint is simplified as:
Figure FDA0002705780630000039
Figure FDA00027057806300000310
Figure FDA00027057806300000311
θp∈θ′side,
Figure FDA00027057806300000312
a subscript P1,.. q.
And solving the second-order cone constraint through an interior point method to obtain the optimal solution of the weighting vector w under the second-order cone constraint.
6. The conformal array two-dimensional beam optimization method based on convex optimization theory as claimed in claim 5, wherein the two-dimensional beam in step (7) is formed by two-dimensional beam response
Figure FDA00027057806300000313
The calculation method is as follows:
Figure FDA00027057806300000314
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