CN115825875A - Robust low sidelobe beam forming method for improving objective function and constraint - Google Patents

Robust low sidelobe beam forming method for improving objective function and constraint Download PDF

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CN115825875A
CN115825875A CN202310110892.XA CN202310110892A CN115825875A CN 115825875 A CN115825875 A CN 115825875A CN 202310110892 A CN202310110892 A CN 202310110892A CN 115825875 A CN115825875 A CN 115825875A
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covariance matrix
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constraint
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CN115825875B (en
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蔡凌萍
邹阳
初瑞雪
李洪涛
田巳睿
邢灵尔
钱浩楠
邱林康
余其旺
黄雪琴
狄儒霄
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Nanjing University of Science and Technology
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Abstract

The invention discloses a method for forming a steady low sidelobe beam by improving a target function and constraint, which is used for sampling a received signal of a radar array; obtaining a desired signal guide vector and a covariance matrix; correcting the array guide vector and the covariance matrix; solving array receiving power based on the corrected covariance matrix; according to the beam width range of the mismatching of the steering vector, the upper bound of the uncertain set is solved
Figure ZY_1
And determining the width of the main lobe; determining side lobe level constraint when the steering vectors are mismatched, and establishing a steady low side lobe beam forming model for improving a target function and the constraint; solving the beam forming model by a convex optimization method to obtain an optimal weight vector; and finally obtaining beam forming output. The invention not only has wider main peak width, but also has larger null width at the interference position than other beam formers, no matter the desired signalThe invention can effectively deal with the mismatching of the guide vector when the signal direction has deviation or the interference direction has deviation, and has obvious low side lobe level.

Description

Robust low sidelobe beam forming method for improving objective function and constraint
Technical Field
The invention belongs to the technical field of beam forming of digital array radars, and particularly relates to a robust low sidelobe beam forming method for improving a target function and constraint.
Background
Beamforming is a common spatial filtering technique in array processing, which by computing an array can extract the desired signal and suppress interference from different directions. Modern beamforming methods are data dependent. The most well-known data-dependent beamforming technique is the minimum variance distortion free response (MVDR) beamformer, which adaptively maximizes the power in the desired direction and the signal to interference and noise ratio of the array output. However, the MVDR beamformer needs to know the steering vector of the desired signal accurately, which is not practical in practical applications. In fact, many factors may affect the accuracy of the steering vector, such as angle-of-arrival estimation errors, array calibration inaccuracies, and antenna position mismatches. The steering vector and the sampling covariance matrix greatly affect the output performance of the beam former, and when the steering vector is mismatched or the covariance matrix is wrong, the performance of the traditional beam forming algorithm is seriously reduced. Meanwhile, the selection of the main lobe width also influences the performance of the algorithm, if the main lobe width is too large, extra noise and interference are introduced, and the output SINR of the algorithm is influenced; if the main lobe width is too small, the desired signal is outside the main peak width, and the output gain of the algorithm to the desired direction signal will decrease. In addition, if the sidelobe level of the beam is too high, target detection performance, low interception performance and interference resistance performance may be reduced.
Many robust beamforming techniques are currently developed, such as signal plus interference subspace component using sample covariance matrix, and a eigenspace-based beamformer is proposed, but it has a drawback: performance decreases with decreasing signal-to-noise ratio or increasing number of interferers. The LCMV beamformer widens the main beam by adding additional linear constraints, but it can only handle angle of arrival (AOA) mismatches. And as more linear constraints are added, the degree of freedom of interference suppression thereof decreases. The subspace beamformer estimates the noise subspace and the interference subspace by constructing an interference-plus-noise covariance matrix, which can achieve high resolution when the Array Steering Vectors (ASVs) are mismatched, but the computational complexity is high, and at low signal-to-noise ratios, the signal subspace may be corrupted by the noise subspace and the performance of the subspace beamformer drops dramatically. Furthermore, it requires that the dimensions of the signal plus interference subspace are known exactly and are much lower than the number of sensors, which means that it requires large snapshots and places stringent requirements on the amount of interference.
The above robust beamforming algorithms have different defects in solving the mismatch problem, and cannot accurately control the sidelobe level of the beam and meet the practical application requirements of robust low-sidelobe beamforming.
Disclosure of Invention
The invention aims to provide a robust low sidelobe beam forming method for improving an objective function and constraint.
The technical solution for realizing the purpose of the invention is as follows: a robust low sidelobe beamforming method for improving objective function and constraints, comprising the steps of:
step 1: signal sampling: acquiring a receiving signal of a radar array, and sampling by using a set snapshot number;
and 2, step: and (3) solving a sampling covariance matrix and an expected signal guide vector: calculating a sampling covariance matrix by using the array receiving data in the step 1, and calculating an expected signal guide vector according to the known expected signal direction of a target;
and 3, step 3: error terms are added to modify the array signal steering vector and the sampling covariance matrix. The corrected array signal guide vector and the sampling covariance matrix are respectively as follows:
Figure SMS_1
wherein ,
Figure SMS_3
steering vectors for any desired array;
Figure SMS_7
for steering vector errors, it is modeled as a spherical region, i.e.
Figure SMS_9
, wherein ,
Figure SMS_2
is the upper bound of the array steering vector uncertainty set;
Figure SMS_5
is an ideal sampling covariance matrix and is,
Figure SMS_8
an uncertain parameter representing a sampling covariance matrix, an
Figure SMS_10
，
Figure SMS_4
The norm of the matrix is represented,
Figure SMS_6
the upper bound of the sampling covariance matrix uncertainty set is defined;
and 4, step 4: to findTaking the array received power after the sampling covariance matrix is corrected: and (4) solving the array received power based on the sampling covariance matrix corrected in the step (3) and improving the target function. Wherein, the array received power, i.e. the improved objective function, is represented as:
Figure SMS_11
，
can be equivalently changed into
Figure SMS_12
，
wherein ,
Figure SMS_13
is a weight vector;
Figure SMS_14
the corrected sampling covariance matrix is obtained;
Figure SMS_15
is an ideal sampling covariance matrix and is,
Figure SMS_16
an uncertainty parameter representing a sampling covariance matrix;
Figure SMS_17
the upper bound of the sampling covariance matrix uncertainty set is defined;
and 5: and (3) solving an upper bound of the array steering vector uncertainty set: according to the beam width range of mismatching of the array steering vector, the upper bound of the uncertain set is obtained
Figure SMS_18
. Wherein, to ensure that the array bias is within the half-power beamwidth, the upper bound of the uncertainty set
Figure SMS_19
The requirements are as follows:
Figure SMS_20
wherein ,
Figure SMS_21
the number of the array elements is the number of the array elements,
Figure SMS_22
the distance between the array elements is the same as the distance between the array elements,
Figure SMS_23
a wavelength at which the array emits signals;
Figure SMS_24
, wherein
Figure SMS_25
Is the desired signal direction;
step 6: determining the width of a main lobe: upper bound based on array steering vector uncertainty set
Figure SMS_26
And the left and right boundaries of the main lobe are obtained by utilizing the symmetry of the spatial domain and the frequency domain processing, so that the width of the main lobe is determined. Wherein, the left and right boundaries of the main lobe are obtained by utilizing the symmetry of the spatial domain and the frequency domain processing as follows:
Figure SMS_27
wherein ,
Figure SMS_30
is at an angle of
Figure SMS_32
Conjugate transpose of corresponding signal steering vectors in time;
Figure SMS_35
and
Figure SMS_29
the left and right boundaries of the side lobe constraint region are respectively;
Figure SMS_33
is a vector of the weights to be used,
Figure SMS_36
is composed of
Figure SMS_38
The conjugate transpose of (1);
Figure SMS_28
is at an angle of
Figure SMS_31
A corresponding signal steering vector of time;
Figure SMS_34
is a static weighting vector;
Figure SMS_37
is a main peak ripple;
and 7: determining side lobe constraints: determining side lobe peak values required by each position of a side lobe region based on actual setting, and determining side lobe level constraint when the guide vector is mismatched by combining a triangular inequality and a norm inequality. Wherein, the side lobe level constraint when the array steering vector is mismatched is as follows:
Figure SMS_41
final equivalent transformation
Figure SMS_44
. wherein ,
Figure SMS_47
to be corrected
Figure SMS_39
An array steering vector of directions;
Figure SMS_45
is composed of
Figure SMS_48
The ideal array of directions leads to a vector,
Figure SMS_50
，
Figure SMS_40
in order to be able to transmit the frequency of the electromagnetic waves,
Figure SMS_43
in order to be the frequency of the signal,
Figure SMS_46
array element spacing;
Figure SMS_49
the peak value of the side lobe required by each position is expressed by dB, and different values can be appointed to different positions according to actual requirements;
Figure SMS_42
a side valve constraint area;
and 8: solving the weight vector: and (3) establishing a robust low sidelobe beam forming model for improving the objective function and the constraint based on the relevant parameters obtained in the steps 1 to 7, and solving the model by using a convex optimization method to obtain a global optimal weight vector. Wherein, the step of solving the weight vector specifically comprises the following steps: solving a steady low side lobe beam forming model for improving an objective function and constraint by a convex optimization method, namely applying a CVX tool box of MATLAB to obtain a weight vector
Figure SMS_51
The robust low sidelobe beamforming model with the improved objective function and the constraint is as follows:
Figure SMS_52
in the formula ,
Figure SMS_55
in order to be the objective function, the target function,
Figure SMS_58
in order to sample the covariance matrix,
Figure SMS_61
the upper bound of the sampling covariance matrix uncertainty set is defined;
Figure SMS_54
the side lobe peak value;
Figure SMS_57
is at an angle of
Figure SMS_60
The corresponding signal in time is directed to a vector,
Figure SMS_63
is at an angle of
Figure SMS_53
Conjugate transpose of corresponding signal steering vectors in time;
Figure SMS_56
is the upper bound of the uncertainty set of the steering vector;
Figure SMS_59
in order to weight the vector statically,
Figure SMS_62
is the main peak corrugation;
and step 9: obtaining a robust low sidelobe adaptive beam: and (4) multiplying the received signal vector obtained in the step (1) with the optimal weight vector obtained in the step (8) to obtain the steady low-sidelobe adaptive beam. The obtained robust low sidelobe adaptive beam is as follows:
Figure SMS_64
wherein ,
Figure SMS_65
for the global optimal weight vector obtained in step 8,
Figure SMS_66
is composed of
Figure SMS_67
The conjugation transpose of (1);
Figure SMS_68
is the received signal vector in step 1.
Compared with the prior art, the invention has the following remarkable advantages:
1) The invention selects reasonable
Figure SMS_69
The main peak performance is better, more expected signal steering vector errors can be tolerated, and the null width at the interference position is larger;
2) The side valve of the invention is low; the invention adds side lobe constraint conditions on the original MVDR beam forming model so as to realize the performance requirement of low side lobe;
3) Interference suppression is good; and solving the weight vector of the model added with the side lobe constraint through a convex optimization method, so that the interference suppression degree is deepened compared with that of a conventional tool.
Drawings
Fig. 1 is a flow chart of the robust low sidelobe beamforming method of the present invention with improved objective function and constraints.
FIG. 2 illustrates the method of the embodiment considering the main peak angles respectively
Figure SMS_70
And is and
Figure SMS_71
main peak width at change.
FIG. 3 illustrates the method of the embodiment considering the main peak angles of
Figure SMS_72
And the width of the main peak when the angle of the main peak changes.
FIG. 4 illustrates an example method for considering an uncertainty set
Figure SMS_73
Graph of SNR loss.
FIG. 5 is a schematic diagram of beam pattern simulation comparing an embodiment method with a conventional method.
FIG. 6 is a second schematic diagram of beam pattern simulation comparing the method of the embodiment with the prior art.
FIG. 7 is a diagram illustrating a simulation of a beam pattern after comparing the method of the embodiment with the prior art and partially enlarging the main peak.
FIG. 8 is a second simulation diagram of the beam pattern after comparing the method of the embodiment with the prior art and partially enlarging the main peak.
FIG. 9 is a diagram illustrating simulation of a beam pattern after the interference is locally amplified in comparison with the prior art.
FIG. 10 is a second simulation diagram of the beam pattern after the interference is locally amplified according to the comparison between the embodiment and the prior art.
Detailed Description
The technical solution in the embodiments of the present invention is further described below with reference to the drawings of the specification.
Referring to fig. 1 to 10, the present embodiment provides a robust low sidelobe beamforming method with improved objective function and constraint, which is used to improve the main peak performance of a beamformer, and meanwhile, the null width at the interference position is also larger than that of the other beamformers, so that the present invention can effectively cope with the mismatching of steering vectors no matter the deviation occurs in the desired signal direction or in the interference direction, and the proposed algorithm has a low sidelobe level that is not possessed by the other beamformers.
As shown in fig. 1, the present invention provides a robust low sidelobe beamforming method for improving objective function and constraint, comprising the steps of:
step 1: signal sampling: the number of array elements of the isotropic uniform linear array is set to be N =32, and the distance between the array antennas is half wavelength. Desired signal direction
Figure SMS_74
(ii) a The signal-to-noise ratio of the desired signal is 20dB, and the interference angle independent of the signal statistics is
Figure SMS_78
The interference-to-noise ratio is 20dB; the angular search interval of the array pattern is
Figure SMS_79
The angle search range is
Figure SMS_76
(ii) a The side valve region is
Figure SMS_77
. Obtaining the receiving signal of the radar array, and sampling with the snapshot number set as 500 to obtain the receiving signal vector
Figure SMS_80
, wherein
Figure SMS_81
Which represents the transpose of the matrix,
Figure SMS_75
(ii) a 500 is the number of sampling points;
step 2: and (3) solving a sampling covariance matrix and a guide vector: utilizing the sampling snapshot data in the step 1 to obtain a sampling covariance matrix
Figure SMS_82
(ii) a According to the known desired signal direction of the target
Figure SMS_83
Calculating to obtain the desired signal steering vector of
Figure SMS_84
；
And step 3: adding an error term to correct the array signal steering vector and the sampling covariance matrix;
specifically, the modified array signal steering vector is:
Figure SMS_85
. wherein ,
Figure SMS_86
steering vectors for any desired array;
Figure SMS_87
uncertain region
Figure SMS_88
Is modeled as a spherical region in which, among other things,
Figure SMS_89
is the upper bound of the array steering vector uncertainty set;
the modified receive covariance matrix is:
Figure SMS_90
. wherein ,
Figure SMS_91
is an ideal sampling covariance matrix and is,
Figure SMS_92
representing uncertainty parameters of the sampled covariance matrix and resembling the definition of an array-oriented vector uncertainty set, constraining it within a known upper bound, an
Figure SMS_93
，
Figure SMS_94
Representing the norm of the matrix.
And 4, step 4: and (3) solving the array received power after the sampling covariance matrix is corrected: and (4) solving the array received power based on the sampling covariance matrix corrected in the step (3) and improving the target function.
Specifically, the array received power can be expressed as:
Figure SMS_95
i.e. by
Figure SMS_96
Consider first:
Figure SMS_97
. The solution of the above formula is
Figure SMS_98
And if and only if
Figure SMS_99
Is obtained by
Figure SMS_100
To represent
Figure SMS_101
The identity matrix of (2). So that the array received power is ultimately converted equivalently
Figure SMS_102
。
And 5: and (3) solving an upper bound of the array steering vector uncertainty set: according to the beam width range of the mismatching of the steering vector, the upper bound of the uncertain set is obtained
Figure SMS_103
；
Specifically, in the main peak direction, when the array weighting vector is as
Figure SMS_104
Then, in the main lobe direction, the normalized gain of the array directional diagram is
Figure SMS_105
（1）
Wherein N is the number of array elements 32, d is the spacing between array elements
Figure SMS_106
。
The directional diagram gain can be regarded as the amplitude response of the spatial filter, and when the main lobe width is obtained, the following equation is obtained
Figure SMS_107
（2）
Order to
Figure SMS_108
Then the equation (2) is equivalently transformed
Figure SMS_109
（3）
Figure SMS_110
An error value representing the desired direction of distance can be obtained
Figure SMS_111
（4）
For an ideal static pattern of uniform equidistant linear arrays, the half-power beam width is:
Figure SMS_112
（5）
to ensure that the array bias is within half-power beamwidth, it is desirable to meet
Figure SMS_113
Namely, it is
Figure SMS_114
Finally, find out
Figure SMS_115
. When the indeterminate set
Figure SMS_116
When the magnitude of the steering vector is not more than 1/3, the mismatching of the steering vector is within the half-power beam width of the main peak, the performance loss is within 3dB, and if the magnitude of the steering vector exceeds 1/3, the steering vector falls outside the half-power beam width of the main peak, and large performance loss is caused to the array output. In this example, will
Figure SMS_117
Set to 0.3.
Step 6: determining the width of a main lobe: array-basedUpper bound of column-oriented vector uncertainty set
Figure SMS_118
And the left and right boundaries of the main lobe are obtained by utilizing the symmetry of the spatial domain and the frequency domain processing, so that the width of the main lobe is determined.
Specifically, in step 4, the corrected array steering vector is:
Figure SMS_119
. Definition according to directional diagram
Figure SMS_120
（6）
The derivation yields:
Figure SMS_121
（7）
in the main lobe region, when
Figure SMS_122
When, the formula (7) is:
Figure SMS_123
（8）
according to the symmetry of signal space domain and frequency domain, then
Figure SMS_124
Value range of
Figure SMS_125
By solving the arcsine function, the real main lobe range of the array can be obtained
Figure SMS_126
。
Simultaneously adding main peak consistency constraint:
Figure SMS_127
（9）
wherein
Figure SMS_128
，
Therefore, the left and right boundaries of the main lobe are determined as:
Figure SMS_129
（10）
the specific parameters are substituted as follows:
Figure SMS_130
（11）
and 7: determining side valve constraint: determining the required level of the side lobe region according to the actual setting as
Figure SMS_131
Determining side lobe level constraint when the steering vector is mismatched, and establishing a steady low side lobe beam forming model for improving the objective function and the constraint based on the related parameters obtained in the steps.
Specifically, the constraint on the side lobe level when the steering vector mismatch is:
Figure SMS_132
（12）
can obtain the product
Figure SMS_133
（13）
According to the definition of vector norm, obviously there are
Figure SMS_134
，
Therefore, equation (12) may be equivalent to the following constraint:
Figure SMS_135
（14）
the following equation is always true for equation (14)
Figure SMS_136
（15）
Thus, the device
Figure SMS_137
（16）
If and only if
Figure SMS_138
The time equal sign holds, wherein
Figure SMS_139
. And is therefore ultimately equivalent to the following constraint
Figure SMS_140
（17）
It turns out that the above equation is a convex function.
In this example, the side lobes are constrained to levels
Figure SMS_141
Set to-20 dB, which is 0.1V.
And 8: solving the weight vector: and (3) establishing a robust low sidelobe beam forming model for improving the objective function and the constraint based on the relevant parameters obtained in the steps 1 to 7, and solving the model by using a convex optimization method to obtain a global optimal weight vector. Wherein, the established improved objective function and constrained robust low sidelobe beam forming model is as follows:
Figure SMS_142
（14）
the specific numerical values substituted into this example are:
Figure SMS_143
（15）
and solving the formula (15) by using a CVX tool box of MATLAB to obtain a global optimal weight vector.
And step 9: obtaining a robust low sidelobe adaptive beam: receiving signal vector obtained in step 1
Figure SMS_144
The optimal weight vector obtained in step 8
Figure SMS_145
And performing multiplication operation to obtain the steady low sidelobe self-adaptive beam. The obtained robust low sidelobe adaptive beam is as follows:
Figure SMS_146
. in the formula ,
Figure SMS_147
is composed of
Figure SMS_148
The conjugate transpose of (1);
the effects of the present invention can be further explained by the following simulation results.
Simulation result 1:
the number of array elements of the isotropic uniform linear array is 32, and the distance between the array antennas is half wavelength. All signals are ideal far-field narrow-band signals. The signal-to-noise ratio of the desired signal is 30dB, and the angle of the interference direction counted independently from the signal is
Figure SMS_149
The interference has a dry to noise ratio of 40dB.
FIG. 2 shows the main peak angles respectively
Figure SMS_150
And is and
Figure SMS_151
main peak width at change; FIG. 3 shows the main peak angles respectively
Figure SMS_152
And the width of the main peak when the angle of the main peak changes. As can be seen from the figure, when
Figure SMS_153
And when the main peak width is within the half-power beam width, the performance loss is small. Figure 4 shows the difference
Figure SMS_154
SNR loss under, wherein the simulated fast beat number is
Figure SMS_155
The Monte Carlo count was 2000. As can be seen from the figures, the,
Figure SMS_156
the SNR loss is within 3 dB.
Simulation result 2:
the number of array elements of the isotropic uniform linear array is 32, and the distance between the array antennas is half wavelength. All signals are ideal far-field narrowband signals. The desired signal direction is
Figure SMS_157
The signal-to-noise ratio of the desired signal is 20dB, and the interference angle independent of the signal statistics is
Figure SMS_162
The interference has a dry to noise ratio of 20dB. The angular search interval of the array pattern is
Figure SMS_165
The angle search range is
Figure SMS_159
The array receiving fast beat number is 500, and the side lobe area is
Figure SMS_160
Error set upper bound set to
Figure SMS_163
Wave of main peak
Figure SMS_166
Upper bound of uncertainty set for array receive covariance matrix
Figure SMS_158
Sidelobe level
Figure SMS_161
Ideal static array weights
Figure SMS_164
。
The simulation realizes the comparison of the method of the embodiment with the existing algorithms, wherein the existing algorithms comprise a static directional diagram, an SMI beam former, an NCCB beam former, a WCPO beam former, an LCMV beam former and a DL beam former. Where the NCCB beamformer norm constraint parameter is set to 0.08, the wcpo beamformer error bound is set to 0.3, and the diagonal loading of the dl beamformer is set to 30. Fig. 5 and 6 show array patterns under various beamformers, and fig. 7 and 8 show the array patterns of various beamformers after local amplification of main peaks. Fig. 9 and 10 show diagrams of various beamformer array patterns after local amplification of interference. It can be seen from the figure that the main lobe width of the invention is maximum, and the null width at the interference position is also larger than that of the rest beam formers, so that the invention can effectively deal with the mismatching of the steering vector no matter the deviation occurs in the expected signal direction or the deviation occurs in the interference direction; at the same time, the present invention has significantly lower sidelobe levels than other beamformers.
In summary, the present invention provides a robust low sidelobe beamforming method for improving objective function and constraint, comprising:
acquiring a receiving signal of a radar array and sampling; obtaining and correcting a guide vector of an output expected signal and a covariance matrix of received data sampling; calculating array receiving power based on the corrected sampling covariance matrix; according to the beam width range of the mismatching of the steering vector, the upper bound of the uncertain set is obtained
Figure SMS_167
(ii) a Based on
Figure SMS_168
Determining the width of a main lobe; determining side lobe level constraint when the steering vectors are mismatched, and establishing a steady low side lobe beam forming model for improving a target function and the constraint; solving the beam forming model by a convex optimization method to obtain an optimal weight vector; and finally obtaining the beam.
The invention not only has wider main peak width, but also the null width at the interference is larger than that of other beam formers, and the invention also has low side lobe level which is not possessed by other beam formers.
The invention is not described in detail, but is well known to those skilled in the art. The foregoing detailed description of the preferred embodiments of the invention has been presented. It should be understood that numerous modifications and variations could be devised by those skilled in the art in light of the present teachings without departing from the inventive concepts. Therefore, the technical solutions available to those skilled in the art through logic analysis, reasoning and limited experiments based on the prior art according to the concept of the present invention should be within the scope of protection defined by the claims.

Claims (8)

1. A method for robust low sidelobe beamforming with improved objective function and constraints, comprising the steps of:
step 1: signal sampling: acquiring a receiving signal of a radar array, and sampling by using a set snapshot number;
step 2: and (3) solving a sampling covariance matrix and an expected signal guide vector: calculating a sampling covariance matrix by using the array receiving data in the step 1, and calculating an expected signal guide vector according to the known expected signal direction of a target;
and step 3: adding error terms to correct the array signal steering vector and the sampling covariance matrix;
and 4, step 4: and (3) solving the array received power after the sampling covariance matrix is corrected: based on the sampling covariance matrix corrected in the step 3, calculating array receiving power and improving a target function;
and 5: and (3) solving an upper bound of the array steering vector uncertainty set: according to the beam width range of mismatching of the array steering vector, the upper bound of the uncertain set is obtained
Figure QLYQS_1
；
Step 6: determining the width of a main lobe: upper bound based on array steering vector uncertainty set
Figure QLYQS_2
The left and right boundaries of the main lobe are obtained by utilizing the symmetry of the spatial domain and the frequency domain processing, so that the width of the main lobe is determined;
and 7: determining side lobe constraints: determining side lobe peak values required by each position of a side lobe region based on actual setting, and determining side lobe level constraints when the guide vectors are mismatched by combining a triangular inequality and a norm inequality;
and 8: solving the weight vector: establishing a robust low sidelobe beam forming model for improving a target function and constraint based on the relevant parameters obtained in the steps 1 to 7, and solving the model by a convex optimization method to obtain a global optimal weight vector;
and step 9: obtaining a robust low sidelobe adaptive beam: and (4) multiplying the received signal vector obtained in the step (1) with the optimal weight vector obtained in the step (8) to obtain the steady low-sidelobe adaptive beam.
2. The method for improved objective function and constrained robust low sidelobe beamforming according to claim 1 wherein the modified steering vector of the array signal and the sampling covariance matrix in step 3 are respectively:
Figure QLYQS_3
，
wherein ,
Figure QLYQS_6
steering vectors for any desired array;
Figure QLYQS_8
for steering vector errors, it is modeled as a spherical region, i.e.
Figure QLYQS_10
, wherein ,
Figure QLYQS_5
is the upper bound of the array steering vector uncertainty set;
Figure QLYQS_9
is an ideal sampling covariance matrix and is,
Figure QLYQS_11
an uncertain parameter representing a sampling covariance matrix, an
Figure QLYQS_12
，
Figure QLYQS_4
The norm of the matrix is represented,
Figure QLYQS_7
the upper bound of the uncertainty set for the sampled covariance matrix.
3. The method for improved objective function and constrained robust low sidelobe beamforming according to claim 1 wherein the array received power in step 4 is represented as:
Figure QLYQS_13
can be equivalently transformed into
Figure QLYQS_14
，
wherein ,
Figure QLYQS_15
is a weight vector;
Figure QLYQS_16
the corrected sampling covariance matrix is obtained;
Figure QLYQS_17
is an ideal sampling covariance matrix and is,
Figure QLYQS_18
an uncertainty parameter representing a sampling covariance matrix;
Figure QLYQS_19
the upper bound of the uncertainty set for the sampled covariance matrix.
4. The method of claim 1 wherein in step 5, to ensure array bias within half-power beamwidth, the upper bound of the uncertainty set is determined
Figure QLYQS_20
The requirements are as follows:
Figure QLYQS_21
，
wherein ,
Figure QLYQS_22
the number of the array elements is the number of the array elements,
Figure QLYQS_23
the distance between the array elements is the same as the distance between the array elements,
Figure QLYQS_24
a wavelength at which the array emits signals;
Figure QLYQS_25
, wherein
Figure QLYQS_26
Is the desired signal direction.
5. The method according to claim 1, wherein in step 6, the left and right boundaries of the main lobe are determined by using the symmetry of spatial domain and frequency domain processing as follows:
Figure QLYQS_27
，
wherein ,
Figure QLYQS_30
is at an angle of
Figure QLYQS_33
Conjugate transpose of corresponding signal steering vectors in time;
Figure QLYQS_36
and
Figure QLYQS_29
respectively as the left and right boundaries of the side lobe constraint region;
Figure QLYQS_31
in the form of a vector of weights,
Figure QLYQS_34
is composed of
Figure QLYQS_37
The conjugate transpose of (1);
Figure QLYQS_28
is at an angle of
Figure QLYQS_32
A corresponding signal steering vector of time;
Figure QLYQS_35
is a static weighting vector;
Figure QLYQS_38
is the main peak ripple.
6. The method for improved objective function and constrained robust low sidelobe beamforming according to claim 1 wherein in step 7 the sidelobe level constraints when the array steering vectors are mismatched are:
Figure QLYQS_39
，
final equivalent transformation
Figure QLYQS_41
, wherein ,
Figure QLYQS_45
to be corrected
Figure QLYQS_48
An array steering vector of directions;
Figure QLYQS_42
is composed of
Figure QLYQS_43
The ideal array of directions leads to a vector,
Figure QLYQS_46
，
Figure QLYQS_49
in order to be able to transmit the frequency of the electromagnetic waves,
Figure QLYQS_40
in order to be the frequency of the signal,
Figure QLYQS_44
array element spacing;
Figure QLYQS_47
the peak value of the side lobe required by each position is expressed by dB, and different values can be appointed to different positions according to actual requirements;
Figure QLYQS_50
the side lobes constrain the region.
7. The method for improving objective function and constrained robust low sidelobe beamforming according to claim 1 wherein the step of solving the weight vector in step 8 is specifically: solving a steady low side lobe beam forming model for improving an objective function and constraint by a convex optimization method, namely applying a CVX tool box of MATLAB to obtain a weight vector
Figure QLYQS_51
The robust low sidelobe beamforming model for improving the objective function and constraining is as follows:
Figure QLYQS_52
，
in the formula ,
Figure QLYQS_54
in order to be the objective function, the target function,
Figure QLYQS_56
in order to sample the covariance matrix,
Figure QLYQS_60
the upper bound of the sampling covariance matrix uncertain set is defined;
Figure QLYQS_55
the side lobe peak value;
Figure QLYQS_57
is at an angle of
Figure QLYQS_59
A corresponding signal steering vector of time;
Figure QLYQS_62
is at an angle of
Figure QLYQS_53
The conjugate transpose of the corresponding signal steering vector in time;
Figure QLYQS_58
is the upper bound of the uncertainty set of the steering vector;
Figure QLYQS_61
in order to weight the vector statically,
Figure QLYQS_63
is the main peak ripple.
8. The method for improving objective function and constraint robust low sidelobe beamforming according to claim 1 wherein the robust low sidelobe adaptive beam obtained in step 9 is:
Figure QLYQS_64
，
wherein ,
Figure QLYQS_65
for the global optimal weight vector obtained in step 8,
Figure QLYQS_66
is composed of
Figure QLYQS_67
The conjugate transpose of (1);
Figure QLYQS_68
is the received signal vector in step 1.
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