阅读说明：本技术 基于直接凸优化建模的鲁棒性自适应波束形成方法 (Robustness self-adaptive beam forming method based on direct convex optimization modeling ) 是由 卓欣然 胡进峰 邹欣颖 周扬 于 2019-11-08 设计创作，主要内容包括：本发明公开了一种基于直接凸优化建模的鲁棒性自适应波束形成算法,属于雷达技术领域。本发明基于其发现的Robust-ADBF存在可减小误差,公开了一种基于直接凸优化建模的鲁棒性自适应波束形成算法,本发明的算法不仅避免了非凸优化问题近似成凸优化带来的误差,还进一步通过信号幅值响应约束来精确控制角度误差范围,并进行交叉迭代求解。本发明在干扰方向凹口更深,与现有波束形成方案相比,具有10dB以上的提升；且输出SINR有较大提升,系统性能更优。(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.)

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

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:

wherein, L represents the angle interval number of scanning, and k represents the weighting coefficient;

ω_na weight vector representing the nth angular sweep;

original variables

matrix array

matrix array

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 toAnd 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:

in cross-iterative solution, define x^k、x^k-1Respectively representing the values of the original variable x before and after the current iteration is updated, y^k、y^k-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 is^k、u^k-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:

u^k＝u^k-1+H(y^k,x^k)。

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:

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: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:

wherein epsilon^priA predetermined threshold, epsilon, of the residual error representing the constraint^dualA preset threshold representing the residual error of the original variable x; epsilon^absRepresenting the absolute error, epsilon, of the steering vector^relDenotes the relative error of the steering vector, and^abs＞0，ε^rel＞0。

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

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:

wherein, L represents the angle interval number of scanning, and k represents the weighting coefficient;

ω_na weight vector representing the nth angular sweep;

original variables

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

Wherein the content of the first and second substances,

θ_nrepresents the source arrival angle, [ theta ] at the nth angular sweep_i，Θ_j]Indicating a preset target angle error range, which is usually obtained based on a priori knowledge, 0 indicating a zero vector, and an intermediate β_nBased on w^Hβ_nw＝|w^Ha(θ_n)|²Determination of a (θ)_n) Is expressed with respect to theta_nThe 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

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:

in cross-iterative solution, define x^k、x^k-1Respectively representing the values of the original variable x before and after the current iteration is updated, y^k、 y^k-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 is^k、u^k-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:

u^k＝u^k-1+H(y^k,x^k)；

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:

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:

and performing cross iterative solution based on L (y, x, z).

The augmented lagrangian function of the objective function can be further described as:

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)

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(θ_D)]∈C^M×DIs a direction matrix, theta ═ theta_S,θ_I]^T∈R^D×1(R represents the real number domain) is the direction of arrival (DOA) vector of the signal source, θ_S＝[θ₁,θ₂,…,θ_s]As vectors of DOA of the desired signal, theta_I＝[θ_s+1,…,θ_D]Is the DOA vector of the interfering signal, a (θ)_m)∈C^M×1A steering vector of an M (M is 1, …, M) th signal source; s (k) C ∈^D×1Are mutually uncorrelated signal vectors; n (k) C^M×1Is a gaussian white noise vector; d represents theta_IDimension 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:

wherein the content of the first and second substances,

is a pre-estimated steering vector, autocorrelation matrix, of the target

x (N) represents the array signal, and N is the number of samples (i.e., fast beat number). The solution yields:

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

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),

Then

Establishing an uncertainty set with norm of maximum mismatch error centered around pre-estimated steering vector as upper bound

ε is the mismatch error upper bound, then the Robust-ADBF problem can be described as:

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

Is approximated to

Approximating the non-convex optimization problem of equation (4) to the convex optimization problem as follows:

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 knowledge_i，Θ_j]And the gain of the signal in the range is ensured, so that the signal gain condition under the ideal condition is obtained:

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 ] A_i，Θ_j]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(θ_n)-α²λ(θ_n)|²＝0,for allθ_n∈Θ (8)

where Θ represents the direction of arrival, α is the optimized scaling parameter,

the optimization problem (4) can then be modeled as:

wherein J (α, w) is the Lagrangian function,

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:

wherein

Wherein, β_nTo re-represent P (theta)_n) An intermediate quantity introduced, i.e. according to | w^Ha(θ_n)|²＝w^Hβ_nw is obtained as β_n；

Thus, the optimization problem of equation (9) is further described as:

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

Converting the non-convex optimization problem into a double-convex optimization problem:

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:

in the ADMM framework, dual variables are introduced for the convenience of solving (14)

The augmented Lagrangian function of equation (14) is:

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:

in the cross iterative solution, the (k +1) th iteration is updated as:

u^k+1:＝u^k+H(y^k+1,x^k+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:

the original residuals are:

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:

ω^k≤ε^priv and v^k≤ε^dual(23)

Wherein:

wherein epsilon^absIndicates 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 ε^abs＞0，ε^rel＞0。

When the iterative update converges, the value of x calculated based on the most recent iteration is calculated according toThe 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

Then, based on the parameters obtained by the solutionAnd 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,ε^abs,ε^rel(ii) a Wherein, due toThat is, the initial value of the parameter x includes α and the initial value of w;

(2) if omega^k≤ε^priV and v^k≤ε^dualThen, iteratively updating y, x and u; otherwise, executing step (4)

u^k+1＝u^k+H(y^k+1,x^k+1)；

(3) When the current iteration update result y is obtained^k+1、x^k+1And u^k+1Then, updating the iteration number k to k +1, and continuing to execute the step (2);

(4) will be current x^kThe 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

Resulting in a final weight vector w.