CN106093878A - A kind of interference noise covariance matrix based on probability constraints reconstruct robust method - Google Patents

Abstract

The invention belongs to Array Signal Processing field, relate generally to interference noise covariance matrix based on the probability constraints reconstruct robust algorithm robustness to interference signal guide Random Vector error.The present invention provides a kind of interference noise covariance matrix based on probability constraints reconstruct robust algorithm (IPNCMR PC), introduce and preset outage probability foundation interference signal guide vector error model based on probability constraints, obtain equivalent random error norm constraint upper limit parameter based on probability constraints, use RCB algorithm that power and the steering vector of interference signal are effectively estimated, improve its estimated accuracy further, obtain interference noise covariance matrix more accurately, thus improve the interference noise covariance matrix restructing algorithm robustness to interference signal guide vector error further.

Description

A Robust Reconstruction Method of Interference Noise Covariance Matrix Based on Probability Constraint

technical field

The invention belongs to the field of array signal processing, and mainly relates to the robustness of the interference noise covariance matrix reconstruction robust algorithm to the random error of the interference signal steering vector based on the probability constraint.

Background technique

The Capon adaptive beamforming algorithm can minimize the output power of the array and maximize the output Signal-to-Interference-plus-Noise Ratio (SINR) while ensuring the undistorted output of the desired signal. It improves the array gain, has better azimuth resolution and stronger interference suppression ability. However, Capon beamforming is based on the assumption that both the desired signal steering vector and the interference noise covariance matrix are known accurately, and it is sensitive to the errors of the desired signal steering vector and the interference noise covariance matrix, and in practical applications , the interference noise covariance matrix is generally difficult to obtain, and is often replaced by the array received data sample covariance matrix. When the number of snapshots received by the array is limited, the performance of the Capon adaptive beamforming algorithm will inevitably decline, especially when the received data of the array contains expected signals, the performance degradation is particularly serious.

In this regard, Gershman et al. proposed a Capon-based Worst-Case Performance Optimization (WCPO) beamforming method in 2003. The core idea is to assume that the real steering vector a(θ ₁ ) of the desired signal is related to the predicted guide vector There is an estimation error between , and the error norm has an upper limit That is, it is assumed that the real steering vector a(θ ₁ ) belongs to the elliptic uncertainty set The design criterion is to maximize the beam output SINR in the worst case, namely is the sample covariance matrix of the data received by the array, and the steering vector obtained by WCPO is denoted as In order to further improve the performance of the robust adaptive beamforming algorithm based on the worst performance optimization, Sergiy A. et al. proposed a robust minimum variance beamforming algorithm based on probability constraints in 2008, introducing a preset outage probability p to Indicates the probability of random error reaching the worst case, adopts a statistical method to replace the deterministic method, establishes a steering vector error model based on probability constraints, and constructs an optimization problem based on probability constraints Therefore, the robustness to the random error of the steering vector of the desired signal is further improved. However, because these two types of algorithms use the sample covariance matrix Instead of the interference-noise covariance matrix R _i+n to calculate the array weights, the sample covariance matrix contains the expected signal components, namely Especially in the case of a limited number of data snapshots received by the array, the real desired signal is mistakenly regarded as an interference signal for null trapping (that is, "self-null trapping"), especially when the desired signal input signal-to-noise ratio SNR is large, so that This causes the array output SINR to gradually deviate from the optimal SINR.

In order to effectively solve this problem, Gu Yujie et al. proposed an interference covariance matrix reconstruction algorithm (Interference-plus-Noise Covariance Matrix Reconstruction, IPNCMR) in 2012. The interference noise covariance matrix is obtained by performing Capon spectral integration on the angle interval of the direction of arrival of the signal, and then based on this matrix, a quadratic constrained quadratic programming problem about the error of the steering vector of the desired signal is established to obtain the beamforming weight, which can greatly improve the Adapt to the performance of the beamforming algorithm. However, the IPNCMR algorithm has some inherent deficiencies. This algorithm needs to accurately know the interference noise structure of the array, that is, the precise steering vector of the interference signal. Estimated by the method of signal-oriented vector estimation. Therefore, the IPNCMR algorithm is sensitive to the error of the steering vector of the interference signal, especially the random error of the steering vector.

In order to improve the robustness of this type of algorithm to the error of the steering vector of the interference signal, Yuan Xiaolei et al. proposed an interference covariance matrix reconstruction algorithm (IPNCMR-WCPO) for any random steering vector based on the WCPO criterion in 2015, similar to Based on the modeling of the steering vector error of the desired signal, an error model based on the worst performance optimization criterion is constructed for the steering vector error of the interference signal d=2,3,...,D, use robust Capon beamforming (Robust Capon Beamforming, RCB) to estimate the power of the dth interfering signal and the steering vector Utilizing the Structural Properties of the Interfering Noise Covariance Matrix to Reconstruct the Interfering Noise Covariance Matrix Considering the Steering Vector Error of the Interfering Signal Improving the robustness of the jamming noise covariance matrix reconstruction algorithm to the jamming signal steering vector error. This algorithm obtains a better output SINR than the IPNCMR algorithm when the expected signal is low input SNR; however, when the input SNR is high, the output SINR still has a certain distance from the optimal output SINR. Therefore, it is very necessary to further study the robust reconstruction algorithm of the jamming noise covariance matrix for the steering vector error of the jamming signal.

Contents of the invention

The object of the present invention is to provide a robust algorithm for interference noise covariance matrix reconstruction based on probability constraints (A Robust Algorithm for Interference-plus-Noise Covariance Matrix Reconstruction Based on Probability Constraints, IPNCMR-PC), which introduces preset interruption probability establishment Based on the probability-constrained interference signal steering vector error model, the probability-based equivalent random error norm constrained upper limit parameter is obtained, and the RCB algorithm is used to effectively estimate the power of the interference signal and the steering vector, further improving its estimation accuracy, and obtaining more Accurate interference noise covariance matrix, so as to further improve the robustness of interference noise covariance matrix reconstruction algorithm to interference signal steering vector error.

The idea of the present invention is: the present invention is based on the structural characteristics of the ideal interference noise covariance matrix is the steering vector of the dth interference signal, d=2,3,...,D, is its power, σ ² is the Gaussian white noise power received by the array, I _N is the N×N identity matrix), firstly, the preset outage probability p _d is introduced to represent the probability that the random error of the steering vector of the d-th interference signal reaches the worst case , to establish a steering vector error model based on probability constraints And assuming that the random error δ _d is a complex symmetric Gaussian random variable with zero mean and variance C _δ-d , the upper limit ε _de of the equivalent random error norm constraints based on probability constraints is obtained. Then the RCB algorithm is used to estimate the power of the dth interfering signal and the steering vector At the same time for the sample covariance matrix Perform eigenvalue decomposition (EVD) to estimate the power of the array receiving Gaussian white noise Thus, the reconstructed interference noise covariance matrix is obtained by using the structural characteristics of the interference noise covariance matrix last use Instead of the sample covariance matrix Probability Constraint Based Steering Vector Error Model for Desired Signals Constructing probability-constrained minimum-variance beamforming optimization problems And assuming that the random error δ ₁ is a complex symmetric Gaussian random variable with zero mean and variance C _δ-1 , the weighted value of the beamforming can be obtained, which can further improve the interference noise covariance matrix reconstruction algorithm for the interference signal steering vector error robustness.

A robust method for reconstructing the interference noise covariance matrix based on probability constraints, the specific steps are as follows:

S1. A uniform linear array composed of M array elements receives D signals from far-field sources, and the directions of arrival of each signal are θ _d ,d=1,...,D, without loss of generality, assuming the first One signal is the desired signal, and the remaining D-1 are interference signals, and assuming that the signals are not correlated with each other, and the signal and noise are not correlated with each other, then the received data of the array under the nth snapshot is recorded as

$\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\underset{\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{\overset{\mathrm{<span\; class="notranslate">\; D.</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>$

Among them, A＝[a(θ ₁ ),...,a(θ _D )] is the flow pattern matrix of the array, s(n) is the signal source vector received by the array, v(n) indicates the noise vector received by the array, Assume it is white Gaussian noise with zero mean. The N snapshot data received by the array can be expressed in the following vector form:

X=[x(1),…,x(N)]=AS+V

S=[s(1),...,s(N)]

V=[v(1),...,v(N)]

The sample covariance matrix of the array received data can be obtained from the array received data matrix X

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; XX</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>$

In general, the real steering vectors of the desired signal and the interference signal are unknown, and are estimated by the corresponding DOA algorithm, which inevitably introduces a certain estimation error. Assuming that the signal d, d=1, 2,..., the estimated steering vector of D is The true signal-steering vector a(θ _d ) is located in the elliptic uncertainty set as follows In d=1,...,D, ε _d represents the pre-estimated steering vector of signal d The upper bound of the norm of the estimation error δ _d between the real steering vector a(θ _d ).

S2, using the sample covariance matrix of the array to receive data to estimate the array received white Gaussian noise power right Perform eigenvalue decomposition (EVD) to obtain its eigenvalues (arranged from large to small) Among them, the D large eigenvalues correspond to the D source signal parts received by the array, and the remaining MD small eigenvalues correspond to the noise parts received by the array, so the noise power can be estimated by the following formula:

S3. Based on the structural characteristics of the ideal interference noise covariance matrix, establish a steering vector error model based on probability constraints for interference signals d, d=2, 3,..., D Obtain the upper limit ε _de of the equivalent random error norm constraint based on the probability constraint, and use the RCB algorithm to estimate the power of D-1 interference signals respectively d=2,...,D and steering vector d=2,...,D.

S31. The pre-estimated steering vector of the interference signal d, d=2, 3,..., D is The true signal-steering vector a(θ _d ) lies in the ellipse uncertainty set In , the outage probability p _d is introduced to represent the probability that the random error of the steering vector of the dth interference signal reaches the worst case, and a steering vector error model based on probability constraints is established Formulate optimization problems based on probabilistic constraints

S32. If it is assumed that the random error δ _d of the steering vector obeys the Gaussian random distribution with zero mean and the covariance matrix is C _δ-d , then the random variable w ^H δ _d obeys the zero mean and the covariance matrix is Gaussian distribution of the random variable w ^H δ _d , assuming that the real and imaginary parts of the random variable w H δ d are statistically independent from each other, then its amplitude |w ^H δ _d | obeys the Rayleigh distribution, thus we can get It can be obtained by a certain transformation Then the optimization problem based on probability constraints can be transformed into By analogy to the original WCPO beamforming optimization problem, we can see that when the covariance matrix is When , the equivalent upper limit of the random error norm constraint is

S33. Using the sample covariance matrix To construct the RCB beamforming optimization problem of the interference signal d:

After a certain arrangement, it is transformed into the following semidefinite programming problem:

$\begin{array}{cc}\underset{\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\end{array}$

Using the existing SeDuMi software or CVX software to solve, the power of the interference signal d can be obtained and the steering vector

S32, take d=2,...,D respectively, and repeat step S31 to obtain the interference signal item in the interference noise covariance matrix At the same time combined with the array received Gaussian white noise power estimated in step S2 The interference noise covariance matrix reconstruction considering the interference signal steering vector error can be obtained

S4. The estimated steering vector of the desired signal is Its true steering vector a(θ ₁ ) is located in the ellipse uncertain set Introduce the outage probability p ₁ to represent the probability that the random error of the steering vector of the desired signal reaches the worst case, and establish a steering vector error model based on probability constraints At the same time, the interference noise covariance matrix estimated in step S3 is used Instead of the sample covariance matrix Construct a probability-constrained minimum-variance beamforming optimization problem:

$\begin{array}{ccc}\underset{\mathrm{<span\; class="notranslate">\; w</span>}}{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; no</span>}}\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; .</span>& \mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; \&delta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; \&le;</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; \&Greater\; Equal;</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\end{array}$

After a certain arrangement, it is transformed into the following second-order cone programming problem:

$\begin{array}{ccc}\underset{\mathrm{<span\; class="notranslate">\; w</span>}}{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; V</span>}\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>& \mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; .</span>& {\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\sqrt{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; l</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; \&delta;</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; |</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; V</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; V</span>}\end{array}$

The existing SeDuMi software or CVX software is used to solve, and its robust array weighted w _IPNCMR-PC is obtained.

The beneficial effects of the present invention are:

First, the preset outage probability is introduced to represent the probability that the random error of the interference signal reaches the worst case, and a statistical method is used to replace the deterministic method, a steering vector error model based on probability constraints is established, and an equivalent random error norm based on probability constraints is obtained Constrain the upper limit parameters, and then use the RCB algorithm to estimate the power of D-1 interfering signals respectively d=2,...,D and steering vector d=2,...,D, to further improve the estimation accuracy and obtain a more accurate interference noise covariance matrix, which can effectively address the inherent shortcomings of the existing interference noise covariance matrix reconstruction and effectively improve the robustness of the beamforming algorithm .

In the S3 step of the present invention, the interference noise covariance matrix is estimated according to the definition of the interference noise covariance matrix, and the steering vector error model based on the probability constraint of all interference signals is established, and the equivalent random error norm constraint upper limit based on the probability constraint is obtained. On this basis, the RCB algorithm is used to estimate the power of all interference signals and their steering vectors, which can further improve the estimation accuracy, obtain a more accurate interference noise covariance matrix, and improve the robustness to the random error of the interference signal steering vector.

Description of drawings

Figure 1 is a flow chart of the method of the present invention.

Fig. 2 is a graph showing the variation of beam output SINR with expected signal input SNR in the present invention.

Fig. 3 is a graph showing the variation of beam output SINR with the number of snapshots received by the array in the present invention.

detailed description

The technical solution of the present invention will be described in detail below in combination with the embodiments and the accompanying drawings.

As shown in Figure 1:

S1. A uniform linear array composed of M array elements receives D signals from far-field sources, and the directions of arrival of each signal are θ _d ,d=1,...,D, without loss of generality, assuming the first One signal is the desired signal, and the remaining D-1 are interference signals, and assuming that the signals are not correlated with each other, and the signal and noise are not correlated with each other, then the received data of the array under the nth snapshot is recorded as

$\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\underset{\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{\overset{\mathrm{<span\; class="notranslate">\; D.</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>$

Among them, A＝[a(θ ₁ ),...,a(θ _D )] is the flow pattern matrix of the array, s(n) is the signal source vector received by the array, v(n) indicates the noise vector received by the array, Assume it is white Gaussian noise with zero mean. The N snapshot data received by the array can be expressed in the following vector form:

X=[x(1),…,x(N)]=AS+V

S=[s(1),...,s(N)]

V=[v(1),…,v(N)]

The sample covariance matrix of the array received data can be obtained from the array received data matrix X

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; XX</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>$

In general, the real steering vectors of the desired signal and the interference signal are unknown, and are estimated by the corresponding DOA algorithm, which inevitably introduces a certain estimation error. Assuming that the signal d, d=1, 2,..., the estimated steering vector of D is The true signal-steering vector a(θ _d ) is located in the elliptic uncertainty set as follows In d=1,...,D, ε _d represents the pre-estimated steering vector of signal d The upper bound of the norm of the estimation error δ _d between the real steering vector a(θ _d ).

S2, using the sample covariance matrix of the array to receive data to estimate the array received white Gaussian noise power right Perform eigenvalue decomposition (EVD) to obtain its eigenvalues (arranged from large to small) Among them, the D large eigenvalues correspond to the D source signal parts received by the array, and the remaining MD small eigenvalues correspond to the noise parts received by the array, so the noise power can be estimated by the following formula:

S3. Based on the structural characteristics of the ideal interference noise covariance matrix, establish a steering vector error model based on probability constraints for interference signals d, d=2, 3,..., D Obtain the upper limit ε _de of the equivalent random error norm constraint based on the probability constraint, and use the RCB algorithm to estimate the power of D-1 interference signals respectively d=2,...,D and steering vector d=2,...,D.

S31. The pre-estimated steering vector of the interference signal d, d=2, 3,..., D is The true signal-steering vector a(θ _d ) lies in the ellipse uncertainty set In , the outage probability p _d is introduced to represent the probability that the random error of the steering vector of the dth interference signal reaches the worst case, and a steering vector error model based on probability constraints is established Formulate optimization problems based on probabilistic constraints

S32. If it is assumed that the random error δ _d of the steering vector obeys the Gaussian random distribution with zero mean and the covariance matrix is C _δ-d , then the random variable w ^H δ _d obeys the zero mean and the covariance matrix is Gaussian distribution of the random variable w ^H δ _d , assuming that the real and imaginary parts of the random variable w H δ d are statistically independent from each other, then its amplitude |w ^H δ _d | obeys the Rayleigh distribution, thus we can get It can be obtained by a certain transformation Then the optimization problem based on probability constraints can be transformed into By analogy to the original WCPO beamforming optimization problem, we can see that when the covariance matrix is When , the equivalent upper limit of the random error norm constraint is

S33. Using the sample covariance matrix To construct the RCB beamforming optimization problem of the interference signal d:

After a certain arrangement, it is transformed into the following semidefinite programming problem:

$\begin{array}{cc}\underset{\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\end{array}$

Using the existing SeDuMi software or CVX software to solve, the power of the interference signal d can be obtained and the steering vector

S32, take d=2,...,D respectively, and repeat step S31 to obtain the interference signal item in the interference noise covariance matrix At the same time combined with the array received Gaussian white noise power estimated in step S2 The interference noise covariance matrix reconstruction considering the interference signal steering vector error can be obtained

S4. The estimated steering vector of the desired signal is Its true steering vector a(θ ₁ ) is located in the ellipse uncertain set Introduce the outage probability p ₁ to represent the probability that the random error of the steering vector of the desired signal reaches the worst case, and establish a steering vector error model based on probability constraints At the same time, the interference noise covariance matrix estimated in step S3 is used Instead of the sample covariance matrix Construct a probability-constrained minimum-variance beamforming optimization problem:

$\begin{array}{ccc}\underset{\mathrm{<span\; class="notranslate">\; w</span>}}{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; no</span>}}\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; .</span>& \mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; \&delta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; \&le;</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; \&Greater\; Equal;</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\end{array}$

After a certain arrangement, it is transformed into the following second-order cone programming problem:

$\begin{array}{ccc}\underset{\mathrm{<span\; class="notranslate">\; w</span>}}{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; V</span>}\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>& \mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; .</span>& {\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\sqrt{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; l</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; \&delta;</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; |</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; V</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; V</span>}\end{array}$

The existing SeDuMi software or CVX software is used to solve, and its robust array weighted w _IPNCMR-PC is obtained.

Embodiment 1,

A uniform linear array composed of 10 array elements receives narrow-band signals emitted by 3 far-field sources. The preset direction of arrival of the desired signal is θ ₁ =5°, and the estimation error of the steering vector is is a zero mean, variance The complex symmetric Gaussian random variable of , whose outage probability is preset to p ₁ . The preset directions of arrival of the two interference signals are θ ₂ =-30° and θ ₃ =40° respectively, then the estimation error of the steering vector is is a zero mean, variance The complex symmetric Gaussian random variable of , the outage probability is preset as p ₂ , p ₃ , and the input signal-to-noise ratio SNR is 30dB. For the desired signal, set σ _δ-1 = 0.3, p ₁ = 0.95, and the input signal-to-noise ratio SNR ranges from -10 to 40dB; for two interference signals, set σ _δ-2 = σ _δ-3 = 0.3, p ₂ =p ₃ =0.95. The number of snapshots received by the array is 200, and 500 Monte Carlo experiments are performed. In each Monte Carlo experiment, the random error of the steering vector of the desired signal and the interference signal can be modeled as

${\mathrm{<span\; class="notranslate">\; \&delta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&xi;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}{\sqrt{\mathrm{<span\; class="notranslate">\; m</span>}}}{<span\; class="notranslate">\; \&lsqb;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\mathrm{<span\; class="notranslate">\; j\&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\mathrm{<span\; class="notranslate">\; j\&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; ...</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\mathrm{<span\; class="notranslate">\; j\&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; \&rsqb;</span>}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}$

Among them, the random variable ξ _d obeys the uniform distribution on the interval [0,σ _δ-d ], and m = 1, 2, ..., the phase of M is a random variable that is uniformly distributed on the interval [0,2π].

details as follows:

①. Obtain the covariance matrix of the array received data from the array received data matrix X Perform EVD on it to get the array received Gaussian white noise power

②. According to the Gaussian distribution of the random error of the steering vector of each interference signal and its interruption probability, calculate the equivalent upper limit of the random error norm constraint as At the same time, using the sample covariance matrix To construct the RCB beamforming optimization problem of the interference signal d, and obtain the power of the interference signal d and the steering vector The reconstruction of the interference noise covariance matrix considering the error of the interference signal steering vector is thus obtained

③. Using the reconstructed interference noise covariance matrix To construct the probability-constrained minimum variance beamforming optimization problem of the desired signal After sorting it out, we get the following second-order cone programming problem The existing SeDuMi software or CVX software is used to solve, and its robust array weighted w _IPNCMR-PC is obtained.

④. Change the SNR of the input signal, repeat ①②③, and obtain the change curve of the output SINR of the interference noise covariance matrix reconstruction robust algorithm based on the probability constraint with the expected signal input SNR SNR.

The IPNCMR-PC weighting design is carried out according to the method of the present invention, and the change curve of the beam output SINR with the expected signal input SNR is obtained as shown in FIG. 2 . In Fig. 2, compare IPNCMR-PC and IPNCMR, IPNCMR-WCPO two kinds of algorithms, can see, utilize the IPNCMR-PC beamforming algorithm that the present invention proposes to output SINR approaching optimal output SINR when low signal-to-noise ratio, far away Better than IPNCMR.

Although with the increase of SNR, the output SINR will gradually deviate from the best output SINR, but it is basically equivalent to the IPNCMR performance; no matter the low SNR or high SNR situation, the performance of the IPNCMR-PC beamforming algorithm proposed by the present invention is all excellent Compared with the IPNCMR-WCPO algorithm, this also verifies that the IPNCMR-PC beamforming algorithm has better robustness to the steering vector error of the interference signal.

Embodiment 2,

A uniform linear array composed of 10 array elements receives narrow-band signals emitted by 3 far-field sources. The preset direction of arrival of the desired signal is θ ₁ =5°, and the estimation error of the steering vector is is a zero mean, variance The complex symmetric Gaussian random variable of , whose outage probability is preset to p ₁ . The preset directions of arrival of the two interference signals are θ ₂ =-30° and θ ₃ =40° respectively, then the estimation error of the steering vector is is a zero mean, variance The complex symmetric Gaussian random variable of , the outage probability is preset as p ₂ , p ₃ , and the input signal-to-noise ratio SNR is 30dB. For the desired signal, set σ _δ-1 = 0.3, p ₁ = 0.95, and the input signal-to-noise ratio SNR ranges from -10 to 40dB; for two interference signals, set σ _δ-2 = σ _δ-3 = 0.3, p ₂ =p ₃ =0.95. The expected signal input SNR is 15dB, the number of snapshots received by the array varies from 100 to 500, and 500 Monte Carlo experiments are performed. In each Monte Carlo experiment, the random error of the steering vector of the desired signal and the interference signal can be modeled as

${\mathrm{<span\; class="notranslate">\; \&delta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&xi;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}}{\sqrt{\mathrm{<span\; class="notranslate">\; m</span>}}}{<span\; class="notranslate">\; \&lsqb;</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\mathrm{<span\; class="notranslate">\; j\&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\mathrm{<span\; class="notranslate">\; j\&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; ...</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{{\mathrm{<span\; class="notranslate">\; j\&phi;</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; d</span>}}}<span\; class="notranslate">\; \&rsqb;</span>}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 3</span>}$

Among them, the random variable ξ _d obeys the uniform distribution on the interval [0,σ _δ-d ], and m = 1, 2, ..., the phase of M is a random variable that is uniformly distributed on the interval [0,2π].

details as follows:

①. Obtain the covariance matrix of the array received data from the array received data matrix X Perform EVD on it to get the array received Gaussian white noise power

②. According to the Gaussian distribution of the random error of the steering vector of each interference signal and its interruption probability, calculate the equivalent upper limit of the random error norm constraint as At the same time, using the sample covariance matrix To construct the RCB beamforming optimization problem of the interference signal d, and obtain the power of the interference signal d and the steering vector The reconstruction of the interference noise covariance matrix considering the error of the interference signal steering vector is thus obtained

③. Using the reconstructed interference noise covariance matrix To construct the probability-constrained minimum variance beamforming optimization problem of the desired signal After sorting it out, we get the following second-order cone programming problem The existing SeDuMi software or CVX software is used to solve, and its robust array weighted w _IPNCMR-PC is obtained.

④. Change the number of array received data snapshots, repeat ①②③, and obtain the change curve of the output SINR of the interference noise covariance matrix reconstruction robust algorithm based on probability constraints with the number of array received data snapshots.

The weighted design of IPNCMR-PC is carried out according to the method of the present invention, and the change curve of the beam output SINR with the number of snapshots received by the array is obtained as shown in FIG. 3 . In Fig. 3, comparing IPNCMR-PC and IPNCMR, IPNCMR-WCPO two kinds of algorithms, utilize the IPNCMR-PC beamforming algorithm that the present invention proposes to output SINR and just reach stability basically when the number of snapshots is few, and the same number of snapshots , the SINR output by INCMR-PC is better than that of IPNCMR and IPNCMR-WCPO, which fully demonstrates the effectiveness of IPNCMR-PC beamforming algorithm.

Claims (1)

1. a kind of interference noise covariance matrix reconstruction robust algorithm based on probability constraint, is characterized in that, comprises the steps: S1. A uniform linear array composed of M array elements receives D signals from far-field sources, and the directions of arrival of each signal are θ _d ,d=1,...,D, without loss of generality, assuming the first One signal is the desired signal, and the remaining D-1 are interference signals, and assuming that the signals are not correlated with each other, and the signal and noise are not correlated with each other, then the received data of the array under the nth snapshot is recorded as $\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\underset{\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{\overset{\mathrm{<span\; class="notranslate">\; D.</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; A</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>$ Among them, A＝[a(θ ₁ ),...,a(θ _D )] is the flow pattern matrix of the array, s(n) is the signal source vector received by the array, v(n) indicates the noise vector received by the array, Assume it is white Gaussian noise with zero mean. The N snapshot data received by the array can be expressed in the following vector form: X=[x(1),…,x(N)]=AS+V S=[s(1),...,s(N)] V=[v(1),...,v(N)] The sample covariance matrix of the array received data can be obtained from the array received data matrix X ${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; XX</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}\underset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>$ In general, the real steering vectors of the desired signal and the interference signal are unknown, and are estimated by the corresponding DOA algorithm, which inevitably introduces a certain estimation error. Assuming that the signal d, d=1, 2,..., the estimated steering vector of D is The true signal-steering vector a(θ _d ) is located in the elliptic uncertainty set as follows ||δ _d || ₂ ≤ε _d },d=1,...,D, where ε _d represents the pre-estimated steering vector of signal d The upper bound of the norm of the estimation error δ _d between the real steering vector a(θ _d ). S2, using the sample covariance matrix of the array to receive data to estimate the array received white Gaussian noise power right Perform eigenvalue decomposition (EVD) to obtain its eigenvalues (arranged from large to small) Among them, the D large eigenvalues correspond to the D source signal parts received by the array, and the remaining MD small eigenvalues correspond to the noise parts received by the array, so the noise power can be estimated by the following formula: S3. Based on the structural characteristics of the ideal interference noise covariance matrix, establish a steering vector error model based on probability constraints for interference signals d, d=2, 3,..., D Obtain the upper limit ε _de of the equivalent random error norm constraint based on the probability constraint, and use the RCB algorithm to estimate the power of D-1 interference signals respectively and the steering vector S31. The pre-estimated steering vector of the interference signal d, d=2, 3,..., D is The true signal-steering vector a(θ _d ) lies in the ellipse uncertainty set In ||δ _d || ₂ ≤ ε _d }, the outage probability p _d is introduced to represent the probability that the random error of the steering vector of the dth interference signal reaches the worst case, and a steering vector error model based on probability constraints is established Formulate optimization problems based on probabilistic constraints S32. If it is assumed that the random error δ _d of the steering vector obeys the Gaussian random distribution with zero mean and the covariance matrix is C _δ-d , then the random variable w ^H δ _d obeys the zero mean and the covariance matrix is Gaussian distribution of the random variable w ^H δ _d , assuming that the real and imaginary parts of the random variable w H δ d are statistically independent from each other, then its amplitude |w ^H δ _d | obeys the Rayleigh distribution, thus we can get It can be obtained by a certain transformation Then the optimization problem based on probability constraints can be transformed into By analogy to the original WCPO beamforming optimization problem, we can see that when the covariance matrix is When , the equivalent upper limit of the random error norm constraint is S33. Using the sample covariance matrix To construct the RCB beamforming optimization problem of the interference signal d: $\begin{array}{cccc}\underset{\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; x</span>}}& {\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}& \mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; .</span>& {\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}_{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; \&Greater\; Equal;</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>{<span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \&le;</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; e</span>}}\end{array}$ After some sorting, it is transformed into the following semidefinite programming problem: $\begin{array}{cc}\underset{\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; min</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\end{array}$ $\begin{array}{cc}\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; .</span>& \left[\begin{array}{cc}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}_{\mathrm{<span\; class="notranslate">\; x</span>}}& \mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>\\ {\mathrm{<span\; class="notranslate">\; a</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>& \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\end{array}\right]<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 0</span>}\end{array}$ $\left[\begin{array}{cc}{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; e</span>}}\mathrm{<span\; class="notranslate">\; I</span>}& \mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span>\\ {<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; h</span>}}& \mathrm{<span\; class="notranslate">\; 1</span>}\end{array}\right]<span\; class="notranslate">\; ></span>\mathrm{<span\; class="notranslate">\; 0</span>}$ Using the existing SeDuMi software or CVX software to solve, the power of the interference signal d can be obtained and the steering vector S32, take d=2,...,D respectively, and repeat step S31 to obtain the interference signal item in the interference noise covariance matrix At the same time combined with the array received Gaussian white noise power estimated in step S2 The interference noise covariance matrix reconstruction considering the interference signal steering vector error can be obtained S4. The estimated steering vector of the desired signal is Its true steering vector a(θ ₁ ) is located in the ellipse uncertain set Introduce the outage probability p ₁ to represent the probability that the random error of the steering vector of the desired signal reaches the worst case, and establish a steering vector error model based on probability constraints At the same time, the interference noise covariance matrix estimated in step S3 is used Instead of the sample covariance matrix Construct a probability-constrained minimum-variance beamforming optimization problem: $\begin{array}{ccc}\underset{\mathrm{<span\; class="notranslate">\; w</span>}}{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; no</span>}}\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; .</span>& \mathrm{<span\; class="notranslate">\; PR</span>}<span\; class="notranslate">\; \{</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}{\mathrm{<span\; class="notranslate">\; \&delta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; \&le;</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; \&Greater\; Equal;</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\end{array}$ After a certain arrangement, it is transformed into the following second-order cone programming problem: $\begin{array}{ccc}\underset{\mathrm{<span\; class="notranslate">\; w</span>}}{\mathrm{<span\; class="notranslate">\; min</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; V</span>}\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>& \mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; .</span>& {\mathrm{<span\; class="notranslate">\; w</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\sqrt{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; l</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; \&delta;</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; |</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ,</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; V</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}\mathrm{<span\; class="notranslate">\; V</span>}\end{array}$ The existing SeDuMi software or CVX software is used to solve, and its robust array weighted w _IPNCMR-PC is obtained.