Robust adaptive beam forming optimization method and system based on waveform estimation and guidance vector uncertainty

Abstract

The invention discloses a robust adaptive beam forming optimization method and system based on waveform estimation and guidance vector uncertainty. The method comprises the following steps: constructing a model for maximizing a signal to interference plus noise ratio of beam forming output; an array snapshot data matrix obtained through sampling is approximated through a limited array snapshot method, the empirical power of the difference between the output of a beam former and a constant modulus signal of interest is calculated, and an approximate estimated value of interference and noise power is obtained; substituting the estimated value into an original model to obtain an approximate estimation model of the signal to interference plus noise ratio; under the model, uncertainty set constraints of mismatching between an actual guide vector and an assumed guide vector are introduced, and a spherical uncertainty set with the assumed guide vector as the center is constructed; therefore, a robust adaptive beam forming optimization problem for maximizing the signal to interference plus noise ratio under the worst condition is formulated, an optimal beam former weight vector and a constant modulus waveform vector are solved, and the robustness of the system can be effectively enhanced.

Description

Technical Field

[0001] The present invention belongs to the field of array signal processing, and more specifically, relates to a robust adaptive beamforming optimization method and system based on waveform estimation and steering vector uncertainty. Background Art

[0002] Adaptive array processing has been extensively studied over the past few decades and has been widely applied in numerous fields, including radar, sonar, communications, and microphone array speech / audio processing. The Capon beamformer is a classic waveform estimation method that adaptively selects a weight vector to minimize its output power, subject to the unit gain constraint in the direction of the signal of interest (SOI), thereby ensuring that the signal of interest is not affected by any distortion. However, due to errors in the direction of arrival (DOA) or array calibration, the SOI steering vector may be inaccurate. In this case, the aforementioned beamformer may treat the SOI as interference and suppress it. To address this issue, robust adaptive beamforming technology has emerged. It is a method that can significantly improve array output performance, such as the signal-to-interference-noise ratio, and is highly effective in dealing with mismatches. Therefore, the study of robust adaptive beamforming is of great significance to this field.

[0003] It is well known that this can lead to significant degradation in beamformer performance due to the mismatch between the actual and assumed steering vectors. Similar types of degradation can also occur when the signal array response is known precisely but the number of training samples is small.

[0004] The invention patent with the prior art publication number CN114726414A proposes a method, system, medium, device and terminal for joint transmission beam optimization. Based on the 5GNR cellular system and densely deployed micro base stations, IRS assistance is introduced to reconstruct the channel and build a heterogeneous network model in the unlicensed frequency band; by jointly optimizing the active beamforming vector and the IRS reflected beamforming vector, useful information is enhanced while offsetting interference, thereby achieving fair transmission for multiple users. This scheme is highly dependent on the accuracy of the channel steering vector, and the performance is prone to drop sharply under directional mismatch. Summary of the Invention

[0005] In order to overcome the problems of the existing beamforming optimization methods in the art, such as strong dependence on the steering vector and insufficient robustness, the present invention provides a robust adaptive beamforming optimization method and system based on waveform estimation and steering vector uncertainty.

[0006] The primary purpose of the present invention is to solve the above technical problems, and the technical solutions of the present invention are as follows:

[0007] A first aspect of the present invention provides a robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty, comprising the following steps:

[0008] Construct a model that maximizes the signal-to-interference-plus-noise ratio of the beamforming output;

[0009] The array snapshot data matrix obtained by sampling is approximated using a finite array snapshot method, and the empirical power of the difference between the beamformer output and the constant modulus signal of interest is calculated to obtain an approximate estimate of the interference plus noise power; the approximate estimate is substituted into a signal to interference plus noise ratio model to obtain an approximate estimate model of the signal to interference plus noise ratio;

[0010] Under the approximate estimation model, the mismatch between the actual guidance vector and the assumed guidance vector is introduced into the uncertainty set constraint, and a spherical uncertainty set is constructed with the assumed guidance vector as the center;

[0011] Combining the approximate estimation model and spherical uncertainty set, a robust adaptive beamforming optimization problem is formulated with the goal of maximizing the worst-case signal-to-interference-plus-noise ratio.

[0012] The optimization problem is solved to obtain an optimal beamformer weight vector and a constant modulus waveform vector.

[0013] Furthermore, the expression for constructing the model for maximizing the signal to interference plus noise ratio of the beamforming output is as follows:

[0014]

[0015] Where α0 is the complex amplitude of the signal of interest, w is the beamformer weight vector, and w H is the conjugate transpose of w, a(θ0) and α(θ i ) are the direction of the signal of interest θ0 and the i-th interference signal θ i The steering vector in the direction, i∈[1,I], θ i is the incident direction of the i-th interference source, I is the number of interference signals, s0(n) and s i (n) are the waveforms of the signal of interest and the i-th interference source at the n-th snapshot, E{·} is the expected operation, which means the statistical averaging of multiple snapshots, σ 2 is the noise power variance, which indicates the intensity of the additive white noise of each sensor.

[0016] Furthermore, the finite array snapshot method is used to approximate the sampled array snapshot data matrix, and the expression for calculating the approximate estimated value E of the interference plus noise power is as follows:

[0017]

[0018] Where X is the matrix of N array snapshots, I is the number of interfering signals, w is the beamformer weight vector, and wH is the conjugate transpose of w, a(θ0) and a(θ i ) are the direction of the signal of interest θ0 and the i-th interference signal θ i The guidance vector in the direction, and are the waveform transposes of the signal of interest and the i-th interference source at all snapshot times, i∈[1,I], I is the number of interference signals, It has a mean of 0 and a variance of σ 2 is the additive white Gaussian noise matrix of independent and identically distributed items, α0 is the complex amplitude of the signal of interest, For the N snapshot matrices received by an antenna array with M sensors, the expression is as follows:

[0019]

[0020] in, is a fixed-modulus signal of interest with directions θ0, α(θ0) and a(θ i ) are the direction of the signal of interest θ0 and the i-th interference signal θ i The guidance vector in the direction, and are the waveform transposes of the signal of interest and the i-th interference source at all snapshot times, It has a mean of 0 and a variance of σ 2 The additive white Gaussian noise matrix of independent and identically distributed items.

[0021] Furthermore, the expression of the approximate estimation model of the signal to interference plus noise ratio is as follows:

[0022]

[0023] Where w is the beamformer weight vector, w H is the conjugate transpose of w, a(θ0) is the steering vector in the direction θ0 of the signal of interest, is the N snapshot matrix received by the antenna array with M sensors, α0 is the complex amplitude of the signal of interest, Transpose the waveform of the signal of interest at all snapshots.

[0024] Furthermore, the robust adaptive beamforming optimization problem with the goal of maximizing the worst-case signal-to-interference-plus-noise ratio is formulated as follows:

[0025]

[0026] Where w is the beamformer weight vector, w His the conjugate transpose of w, α0 is the complex amplitude of the signal of interest, is the constant modulus waveform vector of the signal of interest, and the constraint set N is the number of array snapshots, is the steering vector in the direction of the signal of interest θ0, and the uncertainty set of the steering vector is is an assumed steering vector The sphere centered is M, the number of sensor antenna arrays, and ∈ is the error limit parameter. The above optimization problem is equivalent to the following fractional programming problem:

[0027]

[0028] Furthermore, solving the optimization problem includes the following steps:

[0029] By using Charnes-Cooper transformation combined with the strong duality property of linear cone programming, the optimization problem is equivalently transformed to obtain the quadratic matrix inequality optimization problem.

[0030] The alternating optimization algorithm and semidefinite relaxation technique are used to decouple the variables of the quadratic matrix inequality optimization problem and divide it into several convex subproblems.

[0031] Several convex subproblems are solved to obtain the optimal beamformer weight vector and constant modulus waveform vector.

[0032] Furthermore, the expression of the quadratic matrix inequality problem is as follows:

[0033]

[0034] Where λ1, λ2 are real dual variables, α0 is the complex amplitude of the signal of interest, w is the beamformer weight vector, and w H is the conjugate transpose of w, I is the identity matrix, n is the number of array snapshots, is the assumed signal-of-interest steering vector, for The conjugate transpose of s0(n) is a constant modulus waveform, satisfying |s0(n)|=1, is the waveform transpose of the signal of interest at all snapshots, is the N snapshot matrix received by the antenna array with M sensors, X H is the conjugate transpose of X, and ∈ is the error bound parameter.

[0035] Furthermore, the alternating optimization algorithm and semidefinite relaxation technique are used to decouple the variables of the quadratic matrix inequality optimization problem and divide it into several convex subproblems and solve them, including the following steps:

[0036] Randomly initialize the beamformer weight vector w and the complex amplitude α0 of the signal of interest, and use w and α0 to update the constant modulus waveform vector s0 and the real dual variables λ1 and λ2;

[0037] The constant modulus constraint |s0(n)|=1,n=1,…,N is relaxed to |s0(n)|≤1,n=1,…,N. Therefore, the original problem can be reformulated as a convex optimization problem, as shown below:

[0038]

[0039] Where λ1, λ2 are real dual variables, α0 is the complex amplitude of the signal of interest, w is the beamformer weight vector, and w H is the conjugate transpose of w, I is the identity matrix, N is the number of array snapshots, is the assumed signal-of-interest steering vector, for The conjugate transpose of s0(n) is a constant modulus waveform, satisfying |s0(n)|=1, s0 T is the waveform transpose of the signal of interest at all snapshots, is the N snapshot matrix received by the antenna array with M sensors, X H is the conjugate transpose of X, ∈ is the error bound parameter;

[0040] Use CVX tools to find the optimal solution to the convex optimization problem like Each component of is single-mode, then is the optimal solution to the atomic problem; otherwise, by normalization Right now To obtain an approximate solution;

[0041] Using the current optimal solution to fix {λ1,λ2,s0,α0}, the expression of the sub-problem of optimizing the weight vector w is as follows:

[0042]

[0043] The subproblem is transformed into the following linear matrix inequality problem using the semidefinite relaxation technique, which is expressed as follows:

[0044]

[0045] If the solution of the linear matrix inequality problem W ＊ is of rank one, that is, W ＊ =w ＊ w ＊H , then w ＊ It can be used as the original solution. Otherwise, the rank-one solution is found by iterative method. The expression is as follows:

[0046]

[0047] Among them, tr(·) represents the trace of a matrix, W is the optimization variable, and W k is the solution of the kth iteration, ||·|| F is the Frobenius norm, λ1, λ2 are real dual variables, I is the identity matrix, N is the number of array snapshots, α0 is the complex amplitude of the signal of interest, is the assumed signal-of-interest steering vector, for The conjugate transpose of s0 T is the waveform transpose of the signal of interest at all snapshots, is the N snapshot matrix received by the antenna array with M sensors, X H is the conjugate transpose of X, ∈ is the error bound parameter;

[0048] The matrix inequality constraint is re-transformed using Schur's complement theorem, and the expression is as follows:

[0049]

[0050] in, represents the real part of a complex vector,

[0051] W=ww H

[0052]

[0053] Taking the derivative of inequality (1) with respect to α0 and setting the derivative to 0, we get the solution The expression is as follows:

[0054]

[0055] Using the solution renew

[0056] Repeat the above process until the objective function λ2 no longer changes, and obtain the optimal beamformer weight vector w ＊ and constant modulus waveform vector

[0057] A second aspect of the present invention provides a robust adaptive beamforming optimization system based on waveform estimation and steering vector uncertainty, comprising a memory and a processor, wherein the memory comprises a beamforming optimization method program, and when the beamforming optimization method program is executed by the processor, the steps of a robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty are implemented.

[0058] A third aspect of the present invention provides a computer-readable storage medium, which includes a program based on a beamforming optimization method. When the beamforming optimization method program is executed by a processor, it implements the steps of a robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty.

[0059] Compared with the prior art, the beneficial effects of the technical solution of the present invention are:

[0060] The present invention constructs an effective signal-interference-plus-noise modeling method and combines it with explicit modeling of the uncertainty of the signal-of-interest steering vector to propose a robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty. By jointly designing the beamformer weight vector and the constant modulus waveform vector, while maximizing the beamformer output signal-to-interference-plus-noise ratio (SINR), the impact of directional mismatch and small sample effect on performance is effectively suppressed, significantly enhancing the robustness of the system and possessing good practical value and promotion prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] In order to make the purpose and technical solution of the present invention clearer, the present invention provides the following drawings and descriptions:

[0062] Figure 1 A flowchart of a method provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0063] In order to more clearly understand the above-mentioned objects, features and advantages of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be noted that, in the absence of conflict, the embodiments of the present application and the features therein can be combined with each other.

[0064] In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention may also be implemented in other ways different from those described herein. Therefore, the scope of protection of the present invention is not limited to the specific embodiments disclosed below.

[0065] Example 1:

[0066] The present invention provides a robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty, such as Figure 1 Figure 2 shows a flow chart of a robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty. The specific steps are as follows:

[0067] S1: Build a model that maximizes the signal-to-interference-plus-noise ratio (SINR) of the beamforming output;

[0068] More specifically, the expression for constructing the SINR model that maximizes the beamforming output signal to interference plus noise ratio is as follows:

[0069]

[0070] Where α0 is the complex amplitude of the signal of interest (SOI), w is the beamformer weight vector, and w H is the conjugate transpose of w, a(θ0) and a(θ i ) are the direction of the signal of interest θ0 and the i-th interference signal θ i The steering vector in the direction, i∈[1,I], θ i is the incident direction of the i-th interference source, I is the number of interference signals, s0(n) and s i (n) are the waveforms of the signal of interest (SOI) and the i-th interference source at the n-th snapshot, E{·} is the expected operation, which means statistical averaging of multiple snapshots, σ 2 is the noise power variance, which indicates the intensity of the additive white noise of each sensor.

[0071] S2: Approximate the sampled array snapshot data matrix using a finite array snapshot method, calculate the empirical power of the difference between the beamformer output and the constant modulus signal of interest (SOI), and obtain an approximate estimate of the interference plus noise power; substitute the approximate estimate into a signal-to-interference-plus-noise ratio (SINR) model to obtain an approximate SINR estimation model;

[0072] More specifically, the finite array snapshot method is used to approximate the sampled array snapshot data matrix, and the expression for calculating the approximate estimate E of the interference plus noise power is as follows:

[0073]

[0074] Where X is the matrix of N array snapshots, I is the number of interfering signals, w is the beamformer weight vector, and w H is the conjugate transpose of w, a(θ0) and a(θ i ) are the direction of the signal of interest θ0 and the i-th interference signal θ i The guidance vector in the direction, and are the waveform transposes of the signal of interest and the i-th interference source at all snapshot times, i∈[1,I], I is the number of interference signals, It has a mean of 0 and a variance of σ 2 is the additive white Gaussian noise matrix of independent and identically distributed items, α0 is the complex amplitude of the signal of interest (SOI), For the N snapshot matrices received by an antenna array with M sensors, the expression is as follows:

[0075]

[0076] in, is a signal of interest (SOI) with a fixed magnitude and direction θ0, a(θ0) and a(θ i ) are the direction of the signal of interest θ0 and the i-th interference signal θ i The guidance vector in the direction, and are the waveform transposes of the signal of interest and the i-th interference source at all snapshot times, It has a mean of 0 and a variance of σ 2 The additive white Gaussian noise matrix of independent and identically distributed items.

[0077] The expression of the approximate estimation model of the signal to interference plus noise ratio (SINR) is as follows:

[0078]

[0079] Where w is the beamformer weight vector, w H is the conjugate transpose of w, a(θ0) is the steering vector in the direction θ0 of the signal of interest, is the N snapshot matrix received by the antenna array with M sensors, α0 is the complex amplitude of the signal of interest (SOI), Transpose the waveform of the signal of interest at all snapshots.

[0080] S3: Under the approximate estimation model, the mismatch between the actual guidance vector and the assumed guidance vector is introduced into the uncertainty set constraint, and a spherical uncertainty set with an error bound ∈ is constructed with the assumed guidance vector as the center;

[0081] S4: Combining the approximate estimation model and the spherical uncertainty set, we formulate a robust adaptive beamforming optimization problem with the goal of maximizing the worst-case signal-to-interference-plus-noise ratio.

[0082] More specifically, the robust adaptive beamforming optimization problem with the goal of maximizing the worst-case signal-to-interference-plus-noise ratio is formulated as follows:

[0083]

[0084] Where w is the beamformer weight vector, w H is the conjugate transpose of w, α0 is the complex amplitude of the signal of interest (SOI), is the constant modulus waveform vector of the signal of interest (SOI), the constraint set N is the number of array snapshots, is the steering vector in the direction of the signal of interest θ0, and the uncertainty set of the steering vector is is an assumed steering vector The sphere centered is M, the number of sensor antenna arrays, and ∈ is the error limit parameter. The above optimization problem is equivalent to the following fractional programming problem:

[0085]

[0086] S5: Solve the optimization problem to obtain an optimal beamformer weight vector and a constant modulus waveform vector.

[0087] The specific process is:

[0088] By utilizing Charnes-Cooper transformation and the strong duality property of linear cone programming, the optimization problem is equivalently transformed to obtain the quadratic matrix inequality (QMI) optimization problem.

[0089] More specifically, the expression of the quadratic matrix inequality (QMI) problem is as follows:

[0090]

[0091] Where λ1 and λ2 are real dual variables, α0 is the complex amplitude of the signal of interest (SOI), w is the beamformer weight vector, and w H is the conjugate transpose of w, I is the identity matrix, N is the number of array snapshots, is the assumed signal of interest (SOI) steering vector, for The conjugate transpose of s0(n) is a constant modulus waveform, satisfying |s0(n)|=1, is the waveform transpose of the signal of interest at all snapshots, is the N snapshot matrix received by the antenna array with M sensors, X H is the conjugate transpose of X, ∈ is the error limit parameter, and in this embodiment, the value is 0.1M, where M is the number of antenna arrays of the sensor.

[0092] The alternating optimization (AO) algorithm and semidefinite relaxation (SDR) technique are used to decouple the variables of the quadratic matrix inequality (QMI) optimization problem and divide it into several convex subproblems.

[0093] Several convex subproblems are solved to obtain the optimal beamformer weight vector and constant modulus waveform vector.

[0094] More specifically, the alternating optimization (AO) algorithm and the semidefinite relaxation (SDR) technique are used to decouple the variables of the quadratic matrix inequality (QMI) optimization problem and divide it into several convex subproblems and solve them, including the following steps:

[0095] Randomly initialize the beamformer weight vector w and the complex amplitude α0 of the signal of interest (SOI), and use w and α0 to update the constant modulus waveform vector s0 and the real dual variables λ1 and λ2;

[0096] The constant modulus constraint |s0(n)|=1,n=1,…,N is relaxed to |s0(n)|≤1,n=1,…,N. Therefore, the original problem can be reformulated as a convex optimization problem, as shown below:

[0097]

[0098] Where λ1 and λ2 are real dual variables, α0 is the complex amplitude of the signal of interest (SOI), w is the beamformer weight vector, and w H is the conjugate transpose of w, I is the identity matrix, N is the number of array snapshots, is the assumed signal of interest (SOI) steering vector, for The conjugate transpose of s0(n) is a constant modulus waveform, satisfying |s0(n)|=1, s0 T is the waveform transpose of the signal of interest at all snapshots, is the N snapshot matrix received by the antenna array with M sensors, X H is the conjugate transpose of X, ∈ is the error bound parameter;

[0099] Use CVX tools to find the optimal solution to the convex optimization problem like Each component of is single-mode, then is the optimal solution to the atomic problem; otherwise, by normalization Right now To obtain an approximate solution;

[0100] Using the current optimal solution to fix {λ1,λ2,s0,α0}, the expression of the sub-problem of optimizing the weight vector w is as follows:

[0101]

[0102] The subproblem is transformed into the following linear matrix inequality problem using the semidefinite relaxation (SDR) technique, which is expressed as follows:

[0103]

[0104] If the solution of the linear matrix inequality problem W＊ is of rank one, that is, W ＊ =w ＊ w ＊H , then w ＊ It can be used as the original solution. Otherwise, the rank-one solution is found by iterative method. The expression is as follows:

[0105]

[0106] Among them, tr(·) represents the trace of a matrix, W is the optimization variable, and W k is the solution of the kth iteration, ||·|| F is the Frobenius norm, λ1, λ2 are real dual variables, I is the identity matrix, N is the number of array snapshots, α0 is the complex amplitude of the signal of interest (SOI), is the assumed signal of interest (SOI) steering vector, for The conjugate transpose of s0 T is the waveform transpose of the signal of interest at all snapshots, is the N snapshot matrix received by the antenna array with M sensors, X H is the conjugate transpose of X, ∈ is the error bound parameter;

[0107] The matrix inequality constraint is re-transformed using Schur's complement theorem, and the expression is as follows:

[0108]

[0109] in, represents the real part of a complex vector,

[0110] W=ww H

[0111]

[0112] Taking the derivative of inequality (1) with respect to α0 and setting the derivative to 0, we get the solution The expression is as follows:

[0113]

[0114] Using the solution renew

[0115] Repeat the above process until the objective function λ2 no longer changes. In this embodiment, the error is set to 10 -6 , get the optimal beamformer weight vector w ＊ and constant modulus waveform vector

[0116] The present invention constructs an effective signal-interference-plus-noise modeling method and combines it with explicit modeling of the uncertainty of the signal-of-interest steering vector to propose a robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty. By jointly designing the beamformer weight vector and the constant modulus waveform vector, while maximizing the beamformer output signal-to-interference-plus-noise ratio (SINR), the impact of directional mismatch and small sample effect on performance is effectively suppressed, significantly enhancing the robustness of the system and possessing good practical value and promotion prospects.

[0117] Example 2:

[0118] This embodiment provides a robust adaptive beamforming optimization system based on waveform estimation and steering vector uncertainty, including a memory and a processor. The memory includes a beamforming optimization method program. When the beamforming optimization method program is executed by the processor, it implements the steps of the robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty as described in Example 1.

[0119] Example 3:

[0120] This embodiment provides a computer-readable storage medium, which includes a beamforming optimization method program. When the beamforming optimization method program is executed by a processor, the steps of a robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty as described in Example 1 are implemented.

[0121] Obviously, the above embodiments of the present invention are merely examples for the purpose of clearly illustrating the present invention, and are not intended to limit the embodiments of the present invention. Those skilled in the art will appreciate that other variations or modifications can be made based on the above description. It is not necessary and impossible to enumerate all embodiments here. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention shall be included within the scope of protection of the claims of the present invention.

Claims

1. A robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty, characterized in that: The following steps are involved: Construct a model that maximizes the signal-to-interference-plus-noise ratio of the beamforming output; The array snapshot data matrix obtained by sampling is approximated using a finite array snapshot method, and the empirical power of the difference between the beamformer output and the constant modulus signal of interest is calculated to obtain an approximate estimate of the interference plus noise power; the approximate estimate is substituted into a signal to interference plus noise ratio model to obtain an approximate estimate model of the signal to interference plus noise ratio; Under the approximate estimation model, the mismatch between the actual guidance vector and the assumed guidance vector is introduced into the uncertainty set constraint, and a spherical uncertainty set is constructed with the assumed guidance vector as the center; Combining the approximate estimation model and spherical uncertainty set, a robust adaptive beamforming optimization problem is formulated with the goal of maximizing the worst-case signal-to-interference-plus-noise ratio. The optimization problem is solved to obtain an optimal beamformer weight vector and a constant modulus waveform vector.

2. The robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty according to claim 1, characterized in that: The expression for constructing a model that maximizes the signal to interference plus noise ratio of the beamforming output is as follows: Where α0 is the complex amplitude of the signal of interest, w is the beamformer weight vector, and w H is the conjugate transpose of w, a(θ0) and a(θ i ) are the direction of the signal of interest θ0 and the i-th interference signal θ i The steering vector in the direction, i∈[1,I], θ i is the incident direction of the i-th interference source, I is the number of interference signals, s0(n) and s i (n) are the waveforms of the signal of interest and the i-th interference source at the n-th snapshot, E{·} is the expected operation, which means the statistical averaging of multiple snapshots, σ 2 is the noise power variance, which indicates the intensity of the additive white noise of each sensor.

3. The robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty according to claim 1, characterized in that: The finite array snapshot method is used to approximate the sampled array snapshot data matrix. The expression for the approximate estimated value E of the interference plus noise power is as follows: Where X is the matrix of N array snapshots, I is the number of interfering signals, w is the beamformer weight vector, and w H is the conjugate transpose of w, a(θ0) and a(θ i ) are the direction of the signal of interest θ0 and the i-th interference signal θ i The guidance vector in the direction, and are the waveform transposes of the signal of interest and the i-th interference source at all snapshot times, i∈[1,I], I is the number of interference signals, has a mean of 0 and a variance of σ 2 is the additive white Gaussian noise matrix of independent and identically distributed items, α0 is the complex amplitude of the signal of interest, For the N snapshot matrices received by an antenna array with M sensors, the expression is as follows: in, is a fixed-modulus signal of interest with directions θ0, a(θ0) and a(θ i ) are the direction of the signal of interest θ0 and the i-th interference signal θ i The guidance vector in the direction, and are the waveform transposes of the signal of interest and the i-th interference source at all snapshot times, has a mean of 0 and a variance of σ 2 The additive white Gaussian noise matrix of independent and identically distributed entries.

4. The robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty according to claim 1, wherein: The expression of the approximate estimation model of the signal to interference plus noise ratio is as follows: Where w is the beamformer weight vector, w H is the conjugate transpose of w, a(θ0) is the steering vector in the direction θ0 of the signal of interest, is the N snapshot matrix received by the antenna array with M sensors, α0 is the complex amplitude of the signal of interest, Transpose the waveform of the signal of interest at all snapshots.

5. The robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty according to claim 1, characterized in that: The robust adaptive beamforming optimization problem with the goal of maximizing the worst-case signal-to-interference-plus-noise ratio is formulated as follows: Where w is the beamformer weight vector, w H is the conjugate transpose of w, α0 is the complex amplitude of the signal of interest, is the constant modulus waveform vector of the signal of interest, and the constraint set N is the number of array snapshots, is the steering vector in the direction of the signal of interest θ0, and the uncertainty set of the steering vector is is an assumed steering vector The sphere centered is M, the number of sensor antenna arrays, and ∈ is the error limit parameter. The above optimization problem is equivalent to the following fractional programming problem:

6. The robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty according to claim 1, characterized in that: Solving the optimization problem includes the following steps: By using Charnes-Cooper transformation combined with the strong duality property of linear cone programming, the optimization problem is equivalently transformed to obtain the quadratic matrix inequality optimization problem. The alternating optimization algorithm and semidefinite relaxation technique are used to decouple the variables of the quadratic matrix inequality optimization problem and divide it into several convex subproblems. Several convex subproblems are solved to obtain the optimal beamformer weight vector and constant modulus waveform vector.

7. The robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty according to claim 6, characterized in that: The expression of the quadratic matrix inequality problem is as follows: Where λ1, λ2 are real dual variables, α0 is the complex amplitude of the signal of interest, w is the beamformer weight vector, and w H is the conjugate transpose of w, I is the identity matrix, N is the number of array snapshots, is the assumed signal-of-interest steering vector, for The conjugate transpose of s0(n) is a constant modulus waveform, satisfying |s0(n)|=1, is the waveform transpose of the signal of interest at all snapshots, is the N snapshot matrix received by the antenna array with M sensors, X H is the conjugate transpose of X, and ∈ is the error bound parameter.

8. The robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty according to claim 7, characterized in that: The alternating optimization algorithm and semidefinite relaxation technique are used to decouple the variables of the quadratic matrix inequality optimization problem and divide it into several convex subproblems for solution, including the following steps: Randomly initialize the beamformer weight vector w and the complex amplitude α0 of the signal of interest, and use w and α0 to update the constant modulus waveform vector s0 and the real dual variables λ1 and λ2; The constant modulus constraint |s0(n)|=1,n=1,…,N is relaxed to |s0(n)|≤1,n=1,…,N. Therefore, the original problem can be reformulated as a convex optimization problem, as shown below: Where λ1, λ2 are real dual variables, α0 is the complex amplitude of the signal of interest, w is the beamformer weight vector, and w H is the conjugate transpose of w, I is the identity matrix, N is the number of array snapshots, is the assumed signal-of-interest steering vector, for The conjugate transpose of s0(n) is a constant modulus waveform, satisfying |s0(n)|=1, s0 T is the waveform transpose of the signal of interest at all snapshots, is the N snapshot matrix received by the antenna array with M sensors, X H is the conjugate transpose of X, ∈ is the error bound parameter; Use CVX tools to find the optimal solution to the convex optimization problem like Each component of is single-mode, then is the optimal solution to the atomic problem; otherwise, by normalization Right now To obtain an approximate solution; Using the current optimal solution to fix {λ1,λ2,s0,α0}, the expression of the sub-problem of optimizing the weight vector w is as follows: The subproblem is transformed into the following linear matrix inequality problem using the semidefinite relaxation technique, which is expressed as follows: If the solution of the linear matrix inequality problem W ＊ is of rank one, that is, W ＊ =w ＊ w ＊H , then w ＊ It can be used as the original solution. Otherwise, the rank-one solution is found by iterative method. The expression is as follows: Among them, tr(·) represents the trace of a matrix, W is the optimization variable, and W k is the solution of the kth iteration, ||·|| F is the Frobenius norm, λ1, λ2 are real dual variables, I is the identity matrix, N is the number of array snapshots, α0 is the complex amplitude of the signal of interest, is the assumed signal-of-interest steering vector, for The conjugate transpose of s0 T is the waveform transpose of the signal of interest at all snapshots, is the N snapshot matrix received by the antenna array with M sensors, X H is the conjugate transpose of X, ∈ is the error bound parameter; The matrix inequality constraint is re-transformed using Schur's complement theorem, and the expression is as follows: in, represents the real part of a complex vector, W=ww H Taking the derivative of inequality (1) with respect to α0 and setting the derivative to 0, we get the solution The expression is as follows: Using the solution renew Repeat the above process until the objective function λ2 no longer changes, and obtain the optimal beamformer weight vector w ＊ and constant modulus waveform vector 9. A robust adaptive beamforming optimization system based on waveform estimation and steering vector uncertainty, characterized in that: The system includes: a memory and a processor, wherein the memory includes a robust adaptive beamforming optimization method program based on waveform estimation and uncertainty set. When the robust adaptive beamforming optimization method program based on waveform estimation and uncertainty set is executed by the processor, the steps of the robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty according to any one of claims 1 to 8 are implemented.

10. A computer-readable storage medium, characterized in that The computer-readable storage medium includes a beamforming optimization method program. When the beamforming optimization method program is executed by a processor, the steps of a robust adaptive beamforming optimization method based on waveform estimation and steering vector uncertainty as described in any one of claims 1 to 8 are implemented.

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