CN110895327A - Robustness self-adaptive beam forming method based on direct convex optimization modeling - Google Patents

Robustness self-adaptive beam forming method based on direct convex optimization modeling Download PDF

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CN110895327A
CN110895327A CN201911085093.1A CN201911085093A CN110895327A CN 110895327 A CN110895327 A CN 110895327A CN 201911085093 A CN201911085093 A CN 201911085093A CN 110895327 A CN110895327 A CN 110895327A
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CN110895327B (en
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卓欣然
胡进峰
邹欣颖
周扬
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University of Electronic Science and Technology of China
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Abstract

The invention discloses a robustness self-adaptive beam forming algorithm based on direct convex optimization modeling, and belongs to the technical field of radars. The invention discloses a robustness self-adaptive beam forming algorithm based on direct convex optimization modeling, which can reduce errors based on the found Robust-ADBF, not only avoids the error caused by approximation of a non-convex optimization problem to convex optimization, but also accurately controls the angle error range through signal amplitude response constraint and carries out cross iterative solution. The notch is deeper in the interference direction, and compared with the existing beam forming scheme, the notch has the improvement of more than 10 dB; and the output SINR is greatly improved, and the system performance is better.

Description

Robustness self-adaptive beam forming method based on direct convex optimization modeling
Technical Field
The invention belongs to the technical field of beam forming, and particularly relates to a robustness self-adaptive beam forming method based on direct convex optimization modeling.
Background
The Robust adaptive beamforming method (Robust-ADBF) is an effective adaptive beamforming method under the mismatching of steering vectors, and is widely used in practical engineering. In practical engineering applications, due to environmental uncertainties, mismatch of desired Signal steering vectors caused by antenna array position disturbance and antenna manifold distortion, etc., the performance of conventional beamforming is severely reduced, which may be referred to in documents "Feng Y, Liao G, Xu J, Zhu S, Zeng c," Robust adaptive beam-forming adaptive multiple indirect analysis ", Signal Process, 152(2018), pp.320-330", and "j.qian, z.he, w.zhang, et al," Robust adaptive beam-forming for multiple-input multiple-output radial with spatial filtering ", Signal Process, 143.2018, pp.152-160".
The Robust-ADBF still has good adaptive beamforming performance under the conditions of environment uncertainty, desired signal steering vector mismatch caused by antenna array position disturbance, antenna manifold distortion and the like, so the Robust-ADBF has received wide attention. Currently, the main Robust-ADBF approaches can be broadly divided into three categories: the first type is the diagonal Loading (LSMI) method; the second is a spatial projection method; the third category is the indirect convex optimization method.
In the first category of methods, the effect of mismatch errors is usually reduced by adding a diagonal matrix to the covariance matrix.
In the second method, signal components containing desired signal information are projected to a signal space, a desired signal guide vector is separated from an interference vector, accurate estimation of the signal guide vector is achieved, and mismatch influence caused by disturbance is reduced.
In the third method, the Robust-ADBF is modeled into a non-convex optimization problem, and then is approximately converted into a convex optimization problem, so that the indirect convex optimization solution of the non-convex optimization problem is realized.
The three methods are to model the Robust-ADBF into a non-convex optimization problem, then approximate the non-convex optimization problem into a convex optimization problem, and realize indirect solution of the non-convex optimization problem. However, approximation errors are inevitably brought in the process of approximating the non-convex optimization problem to the convex optimization problem; in most methods, the error range of the angle of the signal cannot be accurately controlled, the prior information of the angle error is difficult to accurately utilize, and additional errors may be introduced due to the mismatch of the angle error range.
Disclosure of Invention
The invention aims to: aiming at the problem that the existing Robust-ADBF method firstly models into a non-convex optimization problem and then approximates into a convex optimization problem to solve and bring approximate errors, the Robust-ADBF direct modeling is proposed to be a double convex optimization problem, so that the errors caused by the approximation of the non-convex optimization problem into the convex optimization are avoided; aiming at the technical problem that the angle error range of a target can not be accurately controlled in the processing process of the existing robustness self-adaptive beam forming method, the robustness self-adaptive beam forming method based on the biconvex optimization is provided, and the angle error range is further accurately controlled by reducing or eliminating an approximation link and responding and restricting through signal amplitude, so that the beam forming performance is further improved.
The robustness self-adaptive beam forming method based on the biconvex optimization comprises the following steps:
step S1: obtaining autocorrelation matrix based on array receiving signal x (k) of radar
Figure BDA0002265150480000021
Wherein, x (k) is a column vector with dimension of M, and M represents the receiving array element number of the radar; n represents the number of snapshots of the array received signal, (-)HRepresents a conjugate transpose of the matrix;
step S2: constructing a biconvex optimization model for obtaining an optimal weight vector:
Figure BDA0002265150480000022
wherein, L represents the angle interval number of scanning, and k represents the weighting coefficient;
ωna weight vector representing the nth angular sweep;
original variables
Figure BDA0002265150480000023
w represents a weight vector, α represents an optimized scaling parameter, y represents an auxiliary variable, and x and y are both dimensionsA column vector of degree M + 1;
matrix array
Figure BDA0002265150480000024
Wherein the content of the first and second substances,
Figure BDA0002265150480000025
θnrepresents the source arrival angle, [ theta ] at the nth angular sweepi，Θj]Indicating a preset target angle error range, which is usually obtained based on a priori knowledge, 0 indicating a zero vector, and an intermediate βnBased on wHβnw＝|wHa(θn)|2Determination of a (θ)n) Is expressed with respect to thetanThe guide vector, sign (·)TRepresentation matrix transposition, (.)HRepresents a conjugate transpose of the matrix;
step S3: performing cross iterative solution on the biconvex optimization model based on a preset convergence condition to obtain a solution result x of the biconvex optimization model;
step S4: according to
Figure BDA0002265150480000026
And obtaining a weight vector w as an optimal weight vector, and taking the result of multiplying the optimal weight vector by the array receiving signal x (k) as the output of beam forming.
Further, in step S3, the convergence condition is: and after iterative updating, the original variable x and the auxiliary variable y, and the residual error of the constraint condition (y-x is 0) and the residual error of the original variable (x) of the biconvex optimization model do not exceed respective preset thresholds.
Further, in step S3, an Alternating Direction Multiplier Method (ADMM) may be used to perform a cross iterative solution on the biconvex optimization model, where the solution process specifically includes:
definition H (y, x) ═ y-x ═ 0;
the objective function J (y, x) is defined as:
Figure BDA0002265150480000031
in cross-iterative solution, define xk、xk-1Respectively representing the values of the original variable x before and after the current iteration is updated, yk、 yk-1Respectively representing the values of the auxiliary variable y before and after the current iteration update; initial values of the original variable x and the auxiliary variable y are preset values;
defining a scaling variable u to be (1/rho) z, wherein rho represents a preset penalty parameter, and z represents a dual variable with the dimension of (M +1) multiplied by 1; and u isk、uk-1Respectively representing the values of the scaling variable u before and after the current iteration is updated, wherein the initial value of the scaling variable u is a preset value;
then the iterative update calculation mode of the original variable x, the auxiliary variable y and the scaling variable u in the cross iterative solution is as follows:
Figure BDA0002265150480000032
Figure BDA0002265150480000033
uk＝uk-1+H(yk,xk)；
based on the values of the original variable x and the auxiliary variable y after the iterative update, the residual error of the constraint condition and the residual error of the original variable that can be obtained at present can be specifically expressed as:
Figure BDA0002265150480000034
further, when the ADMM is used to perform cross iterative solution on the biconvex optimization model, the objective function J (y, x) can be described as an augmented lagrangian function of the objective function:
Figure BDA0002265150480000035
and performing cross iterative solution based on L (y, x, z).
The augmented lagrangian function of the objective function can be further described as:
Figure BDA0002265150480000036
and performing cross iterative solution based on L (y, x, z).
In summary, due to the adoption of the technical scheme, the invention has the beneficial effects that:
the invention further accurately controls the angle error range by reducing or eliminating the approximate links and responding and restricting through the signal amplitude, thereby further improving the system performance. The method directly models the Robust-ADBF into a biconvex optimization problem, and can avoid the error caused by the approximation of a non-convex optimization problem into a convex optimization; meanwhile, the invention further controls the angle error range accurately through signal amplitude response constraint and carries out cross iterative solution, thereby enabling the notch of the invention in the Interference direction to be deeper, greatly improving the output SINR (Signal to Interference plus Noise ratio) and enabling the system performance to be better.
Drawings
FIG. 1 is a comparison between the performance of the system under different SNR when the accurate steering vector is known according to the present invention and the existing three schemes: outputting a graph of SINR variation with input SNR;
fig. 2 is a comparison between the performance of the system under different signal-to-noise ratios when the accurate steering vector is known according to the present invention and the existing three schemes: outputting a graph of the SINR along with the change of the fast beat number N;
fig. 3 is a diagram comparing beam forming of the system under different signal-to-noise ratios when the accurate steering vector is known according to the present invention and the existing three schemes:
fig. 4 is a comparison between the present invention and the existing three schemes in the estimation error of the incoming wave direction of the signal: outputting a graph of SINR variation with input SNR;
fig. 5 is a comparison between the present invention and the existing three schemes for estimating the error in the incoming wave direction of the signal: outputting a graph of the SINR along with the change of the fast beat number N;
fig. 6 is a comparison between the present invention and the existing three schemes for estimating the error in the incoming wave direction of the signal: outputting a graph of SINR (signal to interference plus noise ratio) along with the change of the mismatch angle;
FIG. 7 is a comparison of array geometry errors for the present invention and three prior art schemes, in an example: outputting a graph of SINR variation with input SNR;
FIG. 8 is a comparison of array geometry errors for the present invention and three prior art schemes, in an example: outputting a graph of the SINR along with the change of the fast beat number N;
fig. 9 is a comparison of coherent local scattering errors of the present invention with three prior art schemes: outputting a graph of SINR variation with input SNR;
FIG. 10 is a comparison of coherent local scattering errors for the present invention compared to three prior art schemes: and outputting a graph of the SINR along with the change of the fast beat number N.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the following embodiments and accompanying drawings.
Most of the conventional Robust beam forming (Robust-ADBF) methods model the Robust-ADBF into a non-convex optimization problem first and then solve the non-convex optimization problem by further approximating the non-convex optimization problem, and the angle error range of a target cannot be accurately controlled. Aiming at the technical problem, the invention provides a robustness self-adaptive beam forming method based on biconvex optimization. The method directly models the Robust-ADBF into a biconvex optimization problem, and can avoid the error caused by the approximation of a non-convex optimization problem into convex optimization; the invention further controls the angle error range accurately through signal amplitude response constraint and carries out cross iterative solution, thereby enabling the notch of the invention to be deeper in the interference direction, greatly improving the output SINR and ensuring better system performance.
In the beam forming scheme of the invention, in the receiving and processing process of the array radar, an incident signal is assumed to be a far-field narrow-band signal, a receiving end is an M-element uniform linear array, the array element interval d is adopted, and an array signal x (k) received in the kth snapshot is (k)
Figure BDA0002265150480000051
Figure BDA0002265150480000052
Representing a complex field, superscript M × 1 representing a dimension) can be expressed as:
x(k)＝A(θ)s(k)+n(k) (1)
wherein a (θ) ═ a (θ)1),a(θ2),…,a(θD)]∈CM×DIs a direction matrix, theta ═ thetaS,θI]T∈RD×1(R represents the real number domain) is the direction of arrival (DOA) vector of the signal source, θS＝[θ1,θ2,…,θs]As vectors of DOA of the desired signal, thetaI＝[θs+1,…,θD]Is the DOA vector of the interfering signal, a (θ)m)∈CM×1A steering vector of an M (M is 1, …, M) th signal source; s (k) C ∈D×1Are mutually uncorrelated signal vectors; n (k) CM×1Is a gaussian white noise vector; d represents thetaIDimension of (2), symbol (.)TRepresentation matrix transposition, (.)HRepresenting the conjugate transpose of the matrix.
The conventional Minimum Variance Distortionless Response (MVDR) beamforming method solves the weight vector w by solving the following optimal filter:
Figure BDA0002265150480000053
wherein the content of the first and second substances,
Figure BDA0002265150480000054
is a pre-estimated steering vector, autocorrelation matrix, of the target
Figure BDA0002265150480000055
x (N) represents the array signal, and N is the number of samples (i.e., fast beat number). The solution yields:
Figure BDA0002265150480000056
however, in practical applications, the estimated target steering vector and the target true steering vector a (θ)S) There is often an error: namely, it is
Figure BDA0002265150480000057
Where e is the mismatch error. Steering vector mismatch errors can severely impact the performance of a conventional Capon beamformer.
To simplify the description, let a be a (θ)S),
Figure BDA0002265150480000061
Then
Figure BDA0002265150480000062
Establishing an uncertainty set with norm of maximum mismatch error centered around pre-estimated steering vector as upper bound
Figure BDA0002265150480000063
ε is the mismatch error upper bound, then the Robust-ADBF problem can be described as:
Figure BDA0002265150480000064
the constraint condition (subject to) in equation (5) is a non-linear non-convex condition, so the optimization problem is a non-convex optimization problem, and the non-convex optimization problem needs to be solved by converting the non-convex optimization problem into a convex optimization problem through approximation.
Using the Cauchi inequality
Figure BDA0002265150480000065
Is approximated to
Figure BDA0002265150480000066
Approximating the non-convex optimization problem of equation (4) to the convex optimization problem as follows:
Figure BDA0002265150480000067
equation (5) is a second-order cone-convex optimization problem, which can be solved using an interior point method.
Due to the instability of the actual environment, the pre-estimated guide vector and the real target guide vector have mismatch errors. With respect to steering vectors established by error factors other than in equation (4)Firstly, the invention obtains the error range [ theta ] of the target angle by using the prior knowledgei，Θj]And the gain of the signal in the range is ensured, so that the signal gain condition under the ideal condition is obtained:
Figure BDA0002265150480000068
wherein, P (theta)n) 1 denotes that the signal gain in this direction is kept constant, P (θ)n) 0 denotes a signal suppressing this direction. [ theta ] Ai，Θj]The angle error range determined based on the prior signal is represented, so that the angle error range of the target signal can be accurately controlled according to the prior knowledge, and additional errors are avoided.
The constraint in equation (4) is rewritten as:
|P(θn)-α2λ(θn)|2＝0,for allθn∈Θ (8)
where Θ represents the direction of arrival, α is the optimized scaling parameter,
Figure BDA0002265150480000069
the optimization problem (4) can then be modeled as:
Figure BDA00022651504800000610
wherein J (α, w) is the Lagrangian function,
Figure 1
and kappa is a weighting coefficient and can be adjusted according to specific application scenes; l is the number of angular intervals of the scan.
To transform the above problem into a biconvex optimization problem, the present invention transforms equation (8) into:
Figure BDA0002265150480000072
wherein
Figure BDA0002265150480000073
Wherein, βnTo re-represent P (theta)n) An intermediate quantity introduced, i.e. according to | wHa(θn)|2＝wHβnw is obtained as βn；
Thus, the optimization problem of equation (9) is further described as:
Figure BDA0002265150480000074
it is easy to see that the optimization problem in (11) is a non-convex problem, which is an NP-hard problem, and there is no effective way to solve this problem at present. To this end, the invention further introduces an auxiliary variable
Figure BDA0002265150480000075
Converting the non-convex optimization problem into a double-convex optimization problem:
Figure BDA0002265150480000076
equation (12) is a biconvex optimization problem, and can be solved by cross iteration using an Alternating Direction Multiplier Method (ADMM). The auxiliary variable constraint in equation (12) is rewritten to the following form:
H(y,x)＝y-x＝0 (14)
defining an objective function:
Figure BDA0002265150480000077
in the ADMM framework, dual variables are introduced for the convenience of solving (14)
Figure BDA0002265150480000078
The augmented Lagrangian function of equation (14) is:
Figure BDA0002265150480000079
wherein ρ is a penalty parameter, which can be set according to a specific application scenario.
To simplify equation (15), a scaling variable u ═ 1/ρ) z is defined, and equation (15) can be re-described as:
Figure BDA0002265150480000081
in the cross iterative solution, the (k +1) th iteration is updated as:
Figure BDA0002265150480000082
Figure BDA0002265150480000083
uk+1:＝uk+H(yk+1,xk+1) (20)
wherein I represents an identity matrix.
Through the above operation processing, the constraint condition residual error and the original variable residual error can be used as convergence criteria. Then the constraint residual after the kth iteration is:
Figure BDA0002265150480000084
the original residuals are:
Figure BDA0002265150480000085
when the iterative update converges, the two residuals go to 0. Therefore, in the present embodiment, after the kth iteration update, the following convergence criterion is adopted:
ωk≤εpriv and vk≤εdual(23)
Wherein:
Figure BDA0002265150480000086
wherein epsilonabsIndicates the absolute error, ε, of the steering vector (between the steering vector of the pre-estimated target and the steering vector of the actual target)relDenotes the relative error of the steering vector (between the steering vector of the pre-estimated target and the steering vector of the actual target), and εabs＞0，εrel＞0。
When the iterative update converges, the value of x calculated based on the most recent iteration is calculated according to
Figure BDA0002265150480000091
The final weight vector solution result can be obtained, and then multiplied by the received array signal (array data vector) x (k), and the obtained product is the final output result, i.e. the beam forming result.
Namely, the robust adaptive beamforming method based on the biconvex optimization comprises the following specific implementation steps:
firstly, using ADMM method to cross and iterate to solve
Figure BDA0002265150480000092
Then, based on the parameters obtained by the solution
Figure BDA0002265150480000093
And obtaining the optimal weight vector w of the filter, thereby finishing the beam forming processing.
The specific process of utilizing the ADMM method to perform cross iterative solution is as follows:
(1) initializing variables: y is0,x0,u0,ρ,εabs,εrel(ii) a Wherein, due to
Figure BDA0002265150480000094
That is, the initial value of the parameter x includes α and the initial value of w;
(2) if omegak≤εpriV and vk≤εdualThen, iteratively updating y, x and u; otherwise, executing step (4)
Figure BDA0002265150480000095
Figure BDA0002265150480000096
uk+1＝uk+H(yk+1,xk+1)；
(3) When the current iteration update result y is obtainedk+1、xk+1And uk+1Then, updating the iteration number k to k +1, and continuing to execute the step (2);
(4) will be current xkThe value of (A) is used as the result of the ADMM method cross iteration solution. To obtain a final weight vector w, i.e. based on
Figure BDA0002265150480000097
Resulting in a final weight vector w.
Examples
In order to further verify the performance advantages brought by the beamforming method of the present invention, the beamforming performance of the present invention is compared with the existing three beamforming schemes, wherein the existing three beamforming schemes specifically include:
the first scheme is as follows: SMI beamformers, which may be referred to in particular by the references Yi S., Wu Y., Wang Y. "project-based beamforming-compliant beam-forming with a quadratic constraint", Signal Process,122(2016), pp.65-74;
scheme II: LSMI beamformer, specifically referred to in the references X.Li, D.Wang, X.Ma and Z.Xiong, "Robust adaptive beamforming using iterative variable matrix," in Electronics Letters, vol.54, No.9, pp.546-548,352018 ";
the third scheme is as follows: the Worst-Case beamformer can be referred to in particular by the documents "A.Khabbazibasense, S.A.Vorobyov and A.Hassanien," Robust Adaptive Beamforming Based on SteeringVector Estimation With as Little as Possible of the defective beamformer Information, "in IEEETransactions on Signal Processing, vol.60, No.6, pp.2974-2987, June 2012".
In this embodiment, the angle error range in the beam forming scheme of the present invention is set to θ ═ θ -4 °, θ +4 °]In the LSMI scheme, the diagonal loading factor ξ is empirically taken to be 10 σ2，σ2Is the noise energy and assumes an error factor in the beamformer of 3. In this example, all simulation results were subjected to 100 Monte-Carlo independent experiments.
(1) The exact signal steering vector is known.
In the simulation experiment, the target DOA is made to be 0 degree on the assumption that the real guide vector of the target is accurately obtained. Because the target signal is always present in the training unit data, errors are introduced in the calculation of the interference-plus-noise covariance matrix, affecting system performance.
Fig. 1 is a graph showing the variation of the average output SINR of the system with the input SNR when the signal-to-interference-and-noise ratio INR is 30dB and the number of samples N is 30. Fig. 2 is a graph showing the variation of the average output SINR of the system with the number of input samples when the signal-to-interference-and-noise ratio INR is 30dB and the input signal-to-noise ratio SNR is 0 dB. Fig. 3 shows a beam pattern obtained when the target steering vector is accurately known when the signal-to-interference-and-noise ratio INR is 30dB, the input signal-to-noise ratio SNR is 0dB, and the number of samples N is 30. As can be seen from fig. 1, when the accurate steering vector is known, the beamforming scheme of the present invention not only performs better than other algorithms at low signal-to-noise ratio; and in high signal-to-noise ratio, the limited number of samples causes errors in the estimated sample covariance matrix, which indirectly causes mismatching of the steering vectors, and the beam forming scheme of the invention can also ensure good system performance under the condition. Fig. 2 shows that the beamforming scheme of the present invention can achieve stable gain when the number of samples is small, and fig. 3 has a null higher than that of the rest algorithms by more than 10dB in the interference direction, and has a better interference suppression effect. In summary, the beamforming scheme of the present invention has improved system gain compared to the above three existing beamforming schemes.
(2) And (3) signal incoming wave direction estimation error.
In the experiment, the system performance under the condition of steering vector mismatch caused by the fact that the incoming wave direction of the signal is different from the pre-estimated direction is tested. Assuming that the pre-estimated arrival angle is 0 °, the target actual arrival angle is uniformly distributed in [ θ -4 °, θ +4 ° ], the interference signal DOA is 30 °, and the interference-to-noise ratio is 30 dB. The target true angles of arrival are independent of each other in each iteration of the experiment.
Fig. 4 is a graph showing the variation of the average output SINR of the system with the input SNR when the SNR INR is 30dB and the number of samples N is 30. Fig. 5 is a graph showing the variation of the average output SINR of the system with the number of input samples when the SNR is 30dB and the SNR is 0 dB. Fig. 6 is a graph showing the variation of the output SINR with the mismatch angle when the signal-to-interference-and-noise ratio INR is 30dB, the input signal-to-noise ratio SNR is 0dB, and the number of samples N is 30. It can be seen that the performance of the beamforming scheme of the present invention is better than several other existing beamforming schemes, since the beamforming scheme of the present invention avoids the errors introduced by the approximation, and the accurate control of the angular error range reduces the additional errors. As can be seen from fig. 6, as the mismatch angle increases, the loss of the output signal-to-interference-and-noise ratio of the beamforming scheme of the present invention is not large, which indicates that the present invention has better robustness to the steering vector mismatch caused by the incoming wave direction error.
(3) Array geometric errors.
In the experiment, the influence of mismatching of the guide vectors caused by array geometric errors on the signal-to-interference-and-noise ratio output by the system is tested. When the actual array antennas are calibrated, the distance between the antennas has errors, and the actual array distance and the pre-estimated array distance have deviations, so that the pre-estimated steering vector and the actual steering vector have mismatch, and the system performance is affected. The array geometry error was modeled as a uniform random variable distributed over [ -0.05,0.05] intervals in wavelength.
Fig. 7 is a graph showing the variation of the average output SINR of the system with the input SNR when the SNR INR is 30dB and the number of samples N is 30. Fig. 8 is a graph showing the variation of the average output SINR of the system with the number of input samples when the SNR is 30dB and the SNR is 0 dB. As can be seen from the above diagram, because the beamforming scheme of the present invention avoids approximation errors, direct modeling is performed as a cross iterative solution after the biconvex optimization problem, and the output SINR obtained under the condition of fixed sample number in fig. 7 is improved and approaches to an optimal value compared with other existing beamforming schemes. Therefore, the beam forming scheme of the invention has stronger robustness under the condition of mismatching of the steering vectors caused by array geometric errors, and the system performance is improved to some extent.
(4) Coherent local scattering error.
In this experiment, if the target steering vector is mismatched due to the influence of local scattering effect, the real steering vector is
Figure BDA0002265150480000111
Where a is the direct path return information, b (θ)i) Is coherent scatter path information. Wherein theta isiI is 1,2,3,4 independent of each other in each independent experiment and obeys a uniform random distribution with a mean value of 0 ° and a standard deviation of 1 °.
Figure BDA0002265150480000112
Is a random phase of the path and is in [0,2 π]Are uniformly distributed.
Fig. 9 is a graph showing the variation of the system average output SINR with the input SNR when the signal-to-interference-and-noise ratio INR is 30dB and the number of samples N is 30. Fig. 10 is a graph showing the variation of the average output SINR of the system with the number of input samples when the signal-to-interference-and-noise ratio INR is 30dB and the input signal-to-noise ratio SNR is 0 dB. It can be seen from the figure that the performance of the beamforming scheme of the present invention is similar to that of the existing beamforming schemes under the condition of low signal-to-noise ratio, however, after the signal-to-noise ratio is greater than-10 dB, the performance of the beamforming scheme of the present invention is greatly improved compared with that of the existing beamforming schemes, and it can be seen from fig. 10 that the beamforming scheme of the present invention can obtain stable signal gain under the condition of smaller sample number. Therefore, compared with the existing beam forming scheme, the beam forming scheme of the invention has stronger robustness and better system performance.
In conclusion, the beam forming scheme of the invention has deeper notches in the interference direction, and is improved by more than 10dB compared with the existing beam forming scheme; and the output SINR is greatly improved, and the system performance is better.
While the invention has been described with reference to specific embodiments, any feature disclosed in this specification may be replaced by alternative features serving the same, equivalent or similar purpose, unless expressly stated otherwise; all of the disclosed features, or all of the method or process steps, may be combined in any combination, except mutually exclusive features and/or steps.

Claims (6)

1. The robustness self-adaptive beam forming method based on the biconvex optimization is characterized by comprising the following steps of:
step S1: obtaining autocorrelation matrix based on array receiving signal x (k) of radar
Figure FDA0002265150470000011
Wherein, x (k) is a column vector with dimension of M, and M represents the receiving array element number of the radar; n represents the number of snapshots of the array received signal, (-)HRepresents a conjugate transpose of the matrix;
step S2: constructing a biconvex optimization model for obtaining an optimal weight vector:
Figure FDA0002265150470000012
wherein, L represents the angle interval number of scanning, and k represents the weighting coefficient;
ωna weight vector representing the nth angular sweep;
original variables
Figure FDA0002265150470000013
w represents a weight vector, α represents an optimized scaling parameter, y represents an auxiliary variable, and x and y are both column vectors with dimension M + 1;
matrix array
Figure FDA0002265150470000014
Wherein the content of the first and second substances,
Figure FDA0002265150470000015
θnrepresents the source arrival angle, [ theta ] at the nth angular sweepi，Θj]Indicating a preset target angle error range, which is usually obtained based on a priori knowledge, 0 indicating a zero vector, and an intermediate βnBased on wHβnw＝|wHa(θn)|2Determination of a (θ)n) Is expressed with respect to thetanThe guide vector, sign (·)TRepresenting a matrix transposition;
matrix array
Figure FDA0002265150470000016
Step S3: based on a preset convergence condition, carrying out cross iterative solution on an original variable x and an auxiliary variable y of the biconvex optimization model to obtain a final solution result x;
step S4: according to
Figure FDA0002265150470000017
And obtaining a weight vector w as an optimal weight vector, and taking the result of multiplying the optimal weight vector by the array receiving signal x (k) as the output of beam forming.
2. The method as claimed in claim 1, wherein in step S3, the convergence condition is: based on the original variable x and the auxiliary variable y after iterative updating, the residual error of the constraint condition of the biconvex optimization model and the residual error of the original variable do not exceed respective preset thresholds.
3. The method of claim 2, wherein the cross iterative solution is performed on the biconvex optimization model by using an alternating direction multiplier method, and the specific solution process is as follows:
definition H (y, x) ═ y-x ═ 0;
the objective function J (y, x) is defined as:
Figure FDA0002265150470000021
in cross-iterative solution, define xk、xk-1Respectively representing the values of the original variable x before and after the current iteration is updated, yk、yk-1Respectively representing the values of the auxiliary variable y before and after the current iteration update; initial values of the original variable x and the auxiliary variable y are preset values;
defining a scaling variable u to be (1/rho) z, wherein rho represents a preset penalty parameter, and z represents a dual variable with the dimension of (M +1) multiplied by 1; and u isk、uk-1Respectively representing the values of the scaling variable u before and after the current iteration is updated, wherein the initial value of the scaling variable u is a preset value;
then the iterative update calculation mode of the original variable x, the auxiliary variable y and the scaling variable u in the cross iterative solution is as follows:
Figure FDA0002265150470000022
Figure FDA0002265150470000023
uk＝uk-1+H(yk,xk)。
4. the method of claim 3, wherein in performing the cross-iterative solution, the objective function J (y, x) is described as an augmented Lagrangian function of the objective function:
Figure FDA0002265150470000024
and performing cross iterative solution based on L (y, x, z).
5. The method of claim 3, wherein in performing the cross-iterative solution, the objective function J (y, x) is described as an augmented Lagrangian function of the objective function:
Figure FDA0002265150470000025
and performing cross iterative solution based on L (y, x, z).
6. The method of claim 3, wherein the preset thresholds for the residuals of the constraint and the original variables are set based on the updated values of the original variable x and the auxiliary variable y at each iteration, specifically:
Figure FDA0002265150470000031
Figure FDA0002265150470000032
wherein epsilonpriA predetermined threshold, epsilon, of the residual error representing the constraintdualA preset threshold representing the residual error of the original variable x; epsilonabsRepresenting the absolute error, epsilon, of the steering vectorrelDenotes the relative error of the steering vector, andabs＞0，εrel＞0。
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