A small sample adaptive beamforming method based on covariance matrix reconstruction

Abstract

The present invention discloses a small sample adaptive beamforming method based on covariance matrix reconstruction, which belongs to the field of adaptive beamforming technology. The present invention comprises: reducing the dimension of the sample covariance matrix received by the array and the preset expected steering vector, and constructing the reduced dimension matrix using a standard Gaussian random matrix; using the reduced dimension covariance matrix and the steering vector for spatial power spectrum estimation, and obtaining the Monte Carlo expectation of the spatial power spectrum obtained by reducing the dimension of different Gaussian random matrices to obtain an estimated reduced dimension spatial power spectrum; then performing spectral integration on the estimated new spatial power spectrum on the noise sector and the interference sector respectively, and obtaining the noise covariance matrix and the interference covariance matrix respectively, and adding them to obtain the interference plus noise covariance matrix, and then obtaining the expected steering vector based on the matrix; finally, calculating the adaptive beamforming weight vector. The present invention has the advantages of higher signal-to-interference-noise ratio of the beamforming output and no distortion of the expected signal in the case of small samples.

Description

Technical Field

[0001] The present invention belongs to the technical field of adaptive beamforming in array signal processing, and particularly relates to a method for realizing small sample adaptive beamforming by utilizing dimensionality reduction space power spectrum estimation and covariance matrix reconstruction. Background Art

[0002] Adaptive beamforming is a spatial filtering algorithm used in array signal processing to receive signals of interest and suppress interfering signals. It is widely used in radar, sonar, radio astronomy, and wireless communications.

[0003] However, in practical scenarios, adaptive beamforming is sensitive to steering vector errors and covariance matrix errors. Robust adaptive beamforming (RAB) techniques to address these errors have been developed over the past few decades. Diagonal loading is a widely used RAB technique derived from regularization to address the minimum variance distortion-free response problem, but the diagonal loading factor is difficult to select in practical scenarios. The steering vector uncertainty set method can also be considered a type of diagonal loading method. This type of method uses convex optimization to estimate the actual signal steering vector by imposing constraints on the error between the hypothesized steering vector and the actual steering vector. Ledoit and Wolf's method automatically selects the diagonal loading factor (for details, see L. Du, J. Li, and P. Stoica, “Fully automatic computation of diagonal loading levels for robust adaptive beamforming,” IEEE Transactions on Aerospace and Electronic Systems, vol. 46, no. 1, pp. 449–458, Jan. 2010) and applies this technique to RAB to improve performance in small sample cases. The eigenvalue subspace-based method (ESB) eliminates the error part by projecting the hypothetical steering vector into the signal plus interference subspace. However, the dimensionality selection of the signal subspace in this algorithm is a key issue that needs to be solved under low signal-to-noise ratio, which leads to poor performance of the ESB method under low signal-to-noise ratio (SNR) or large steering vector error. If the snapshot contains the desired signal component, the covariance matrix mismatch will lead to the self-zero phenomenon of the desired signal and a significant decrease in performance. At present, the RAB algorithm based on interference plus noise covariance matrix reconstruction has attracted much attention. In the existing processing method, the interference plus noise covariance matrix is ​​reconstructed based on the Capon spectrum by integrating over an angular sector separated from the desired signal direction.

[0004] With the development of array signal processing, the size of arrays is constantly increasing. The prevalence of low-cost sensors means that as the dimension of the covariance matrix increases, the coherence time does not change, so the number of snapshots is usually close to or less than the number of elements. In the case of small sample support, the performance of adaptive beamforming will be seriously deteriorated, especially when the number of snapshots is less than the number of array elements. At this time, the probability that the data covariance matrix is ​​a non-singular matrix is ​​no longer 1. In this case, the Capon spectrum obtained by inverting the sample matrix will no longer be reliable as an integral weight or a criterion for searching for spectral peaks. Few studies have focused on methods for removing the SOI component of the data covariance matrix in the case of small samples. Therefore, in order to ensure the performance of the array output under small sample conditions, it is very necessary to consider how to remove the desired signal component in the covariance matrix and study robust adaptive beamforming algorithms under small sample conditions. Summary of the Invention

[0005] The purpose of the present invention is to overcome the problem that the covariance matrix of existing robust adaptive beamforming algorithms is difficult to invert and the performance is seriously degraded in the case of small samples. The present invention proposes a small sample adaptive beamforming method based on covariance matrix reconstruction. The method can simultaneously combat systematic errors in the case of small snapshots to achieve the purpose of improving the signal-to-interference-noise ratio of the array output.

[0006] The technical solution adopted in the present invention is:

[0007] A small sample adaptive beamforming method based on covariance matrix reconstruction, the method comprising the following steps:

[0008] Step 1: Setting an antenna array, wherein the antenna array is a uniform linear array including M array elements, the first array element of the antenna array is a phase reference point, and the array element spacing is d=λ / 2, where λ represents the signal wavelength;

[0009] Define K (K>1) to represent the number of signal sources, where K signal sources include one desired signal and K-1 interference signals, and define the array incident angle of each signal as θ k , k=0,1,...,K-1;

[0010] Define Θ s represents the angular sector of the desired signal, Θ i represents the angular sector of the interference signal, Θ n represents the noise sector, X(n) represents the array observation data vector;

[0011] Step 2, Capon spectrum estimation in dimensionality reduction space:

[0012] (2a) Calculate the covariance matrix of the array observation data vector X(n) at snapshot number N

[0013] (2b) Construct a random dimensionality reduction matrix based on the array structure:

[0014]

[0015] Among them, the Householder matrix

[0016] Z represents a preset Gaussian random matrix, which obeys the distribution That is, each element is an independent Gaussian random variable with mean 0 and variance 1, I M-1 , I L-1 Represents the M-1 and L-1 dimensional unit matrices respectively; 0 M-1 , 0 L-1 denote the 0 vectors of M-1 and L-1 dimensions respectively; d(θ) denotes the guiding vector in direction θ, I is the M-dimensional identity matrix, and e1 is a unit vector whose first element is 1 and the rest are 0, i.e., e1 = [1, 0, ..., 0] H , L represents the preset target dimension of dimensionality reduction, and L≤min(M,N).

[0017] (2c) Use the random matrix Φ constructed in step 2b to calculate the covariance matrix The dimension reduction is performed with the steering vector d(θ), and then the spatial power spectrum (Capon spectrum) is estimated. The obtained spatial power spectrum estimation result is:

[0018]

[0019] in, The variable is represented by the mean of Z. In the present invention, a method of simulating and sampling Gaussian random matrices is adopted to obtain the Monte Carlo expectation.

[0020] Step 3: Reconstruct the interference plus noise covariance matrix:

[0021] (3a) By n The noise power and noise covariance matrix are calculated by performing Capon spectrum integration; among them, the noise power for: The noise covariance matrix is l(Θ n ) represents the size of the noise sector, that is, the central angle corresponding to the noise sector;

[0022] (3b) The interference covariance matrix is ​​obtained by performing spectral integration in the interference angle sector and then combined with Add together to get the estimated noise plus interference covariance matrix

[0023]

[0024] Step 4: Estimate the desired signal steering vector

[0025] (4a) Modeling Steering Vector: Definition represents the preset steering vector (i.e. the preset steering vector of the desired signal ), e ⊥ Indicates the steering vector error, the preset steering vector and the steering vector error e ⊥ The sum of them is the corresponding true steering vector;

[0026] (4b) Correcting the preset steering vector To maximize the signal-to-interference-plus-noise ratio (SINR) of the beamformer output, solve the following quadratically constrained quadratic programming (QCQP) problem:

[0027]

[0028]

[0029]

[0030] This problem is a convex problem, and after solving it, we can get the estimated value of the true steering vector That is, based on the steering vector error e obtained by the solution ⊥ , and compare it with Add up to get the estimated value

[0031] Step 5: Calculate the adaptive beamforming weight vector:

[0032] (5a) The adaptive beam weight vector is the interference plus noise covariance matrix reconstructed in step 3 and the estimated value obtained in step 4. The calculation shows that:

[0033]

[0034] Therefore, based on the adaptive beamforming weight vector w, the array output of each snapshot can be obtained as y(n)=w(n)X(n).

[0035] This paper proposes using a reduced-dimensional Capon spectrum to reconstruct the interference plus noise covariance matrix, effectively addressing the problem of samples containing desired signal components. First, an improved random reduced-dimensional Capon spectrum calculation method is used, in which a standard random Gaussian matrix is ​​used instead of a Haar matrix to construct the random reduced-dimensional matrix and calculate a new Capon spectrum. The improved Capon spectrum is then used to reconstruct the interference plus noise covariance matrix for the interference and noise angular sectors. Finally, the actual desired signal steering vector is estimated by solving a quadratic programming quadratic constraint problem. The estimated interference plus noise covariance matrix and steering vector are used to solve the beamforming weight vector.

[0036] In summary, the technical solution provided by the present invention brings at least the following beneficial effects:

[0037] The present invention constructs a new robust adaptive beamforming method based on the reconstruction of the interference plus noise covariance matrix, in which the random dimension reduction Capon spectrum estimation used for reconstruction effectively solves the problem that the data covariance matrix is ​​difficult to invert under small samples. At the same time, the present invention only needs to reasonably set the dimension of dimensionality reduction according to the number of snapshots and the number of array elements, avoiding the problem of difficulty in selecting diagonal loading factors in various scenarios in the diagonal loading algorithm, and also avoiding the problem of difficulty in determining the dimensions of the signal subspace and the noise subspace in the characteristic subspace algorithm. Moreover, the reconstructed interference plus noise covariance matrix under small samples can better calibrate the expected signal steering vector affected by various system errors when used for the estimation of the expected signal steering vector. Compared with most current algorithms, the present invention has better performance in error scenarios under small snapshot numbers. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for use in the description of the embodiments. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0039] Figure 1 Schematic diagram of the array arrangement of the present invention.

[0040] Figure 2 Schematic diagram of the relationship between the signal-to-interference-and-noise ratio (SIR) of the array output and SNR in a simulation experiment under random steering vector error according to an embodiment of the present invention.

[0041] Figure 3 Schematic diagram of the relationship between the signal-to-interference-and-noise ratio (SINR) of the array output and the number of snapshots in a simulation experiment under random steering vector error according to an embodiment of the present invention.

[0042] Figure 4 Schematic diagram of the relationship between the signal-to-interference-noise ratio (SIR) and the SNR of the array output in a simulation experiment under incoherent local scattering according to an embodiment of the present invention.

[0043] Figure 5 Schematic diagram of the relationship between the signal-to-interference-noise ratio (SINR) of the array output and the number of snapshots in a simulation experiment under incoherent local scattering according to an embodiment of the present invention. DETAILED DESCRIPTION

[0044] To make the objectives, technical solutions and advantages of the present invention more clear, the embodiments of the present invention will be described in further detail below with reference to the accompanying drawings.

[0045] To address the technical problem of the inability of existing robust adaptive beamforming algorithms to effectively remove the desired signal component in the covariance matrix under small sample conditions, an embodiment of the present invention provides a small sample adaptive beamforming method based on covariance matrix reconstruction. The method first reduces the dimension of the sample covariance matrix received by the array and the preset desired steering vector, and the reduced dimension matrix is ​​constructed using a standard Gaussian random matrix. Then, the reduced dimension covariance matrix and steering vector are used for spatial power spectrum estimation, and the Monte Carlo expectation is calculated for the spatial power spectrum obtained by reducing the dimension of different Gaussian random matrices to obtain an estimated reduced dimension spatial power spectrum. The estimated new spatial power spectrum is then spectrally integrated over the noise sector and the interference sector to obtain a noise covariance matrix and an interference covariance matrix, respectively. The interference plus noise covariance matrix is ​​added together to obtain an interference plus noise covariance matrix. The obtained interference plus noise covariance matrix is ​​used to solve a more accurate desired steering vector in a QCQP problem. Finally, an adaptive beamforming weight vector is calculated. The present invention has the advantages of higher signal-to-interference-noise ratio of the beamforming output and no distortion of the desired signal under small sample conditions.

[0046] As a possible implementation, an embodiment of the present invention provides a small sample adaptive beamforming method based on covariance matrix reconstruction, which specifically includes the following steps:

[0047] Step 1: Set up the antenna array:

[0048] Set up a Figure 1 The one-dimensional uniform linear array shown has a total of M array elements, an array spacing d = λ2, and a narrowband signal frequency of 3.1*10^9 Hz. The array receives K narrowband stationary independent plane wave signals with a wavelength of λ. In this specific embodiment, M = 20, and there are two signal sources K = 3, of which the first signal source is the desired signal source with parameter θ1 = 2°; the other two signal sources are interference signal sources with parameters θ2 = -22° and θ3 = 15°. The spatial noise is zero-mean additive white Gaussian noise, and the noise is independent of the source signal. The desired signal is located in the angular sector Θ s =[θ0-3°,θ0+3°], the interference signal is located at Θ i =[θ1-3°,θ1+3°]∪[θ2-3°,θ2+3°], the pure noise sector is Θ n .

[0049] The steering vector of this uniform linear array is expressed as:

[0050] a(θ)=[1,e j(2πd / λ)sinθ ,...,e j(M-1)(2πd / λ)sinθ ]

[0051] Here, θ represents the incident angle of the signal (source signal).

[0052] The theoretical covariance matrix R can be written as

[0053]

[0054] Among them, σ k represents the power of the kth signal, σ n represents the white noise power, θ k represents the incident angle of the kth signal, and I represents the M-dimensional identity matrix.

[0055] The n snapshot data of the mth array element can be expressed as:

[0056] X(n)=AS(n)+N(n)

[0057] Where X(n)=[x0(n),…,x M (n)] T is the observation data vector, x m (n) represents the observation data of the mi(m=0,1,…,M-1)th array element, S(n)=[s1(n),…,s K (n)] T is the signal source data vector, s k (n) represents the signal source data of the kth (k=1,…,K) signal, A=[a(θ1),…,a(θ K )] is the array flow matrix, a(θ k )=[a 0,k ,…,a M,k ] T is the steering vector, N(n)=[n0(n),…,n M (n)] T is the noise vector.

[0058] Step 2: Capon spectrum estimation in reduced dimensional space:

[0059] In actual work scenarios, the covariance matrix of the observed data X(n) is The calculation formula is:

[0060]

[0061] Wherein, N is the number of snapshots. In this embodiment, N is set to 20.

[0062] Construct the Householder matrix according to the parameters of the uniform linear array:

[0063]

[0064] Here, e1 is a unit vector with the first element being 1 and the rest being 0, and I is the M-dimensional unit matrix. d(θ) represents the steering vector in the corresponding direction. Here, the dimension reduction factor L is selected to ensure that L ≤ min(M,N). For example, for an array with M = 20 and the number of accepted snapshots is 20, the dimension reduction factor L can be set to 10. Sample one hundred standard Gaussian random matrices Z1...Z 100 According to the following construction formula:

[0065]

[0066] Among them, 0 M-1 is an M-1 dimensional all-zero column vector. Construct 100 random dimensionality reduction matrices Φ1...Φ 100 Used for dimensionality reduction Capon spectrum calculation, and to obtain Monte Carlo expectations for the values ​​calculated for different dimensionality reduction matrices:

[0067]

[0068] Step 3: Reconstruct the interference plus noise covariance matrix:

[0069] First, by n The noise power and noise covariance matrix are calculated by using the Capon spectrum integration method. The sampling points in each angular sector are set to 2500. Noise power The calculation method is:

[0070]

[0071] The interference covariance matrix is ​​obtained by performing spectral integration in the interference angle sector, and then added to the noise covariance matrix estimated in step 2 to obtain the estimated noise plus interference covariance matrix Expressed as:

[0072]

[0073] in, represents the power estimate of the rth angular sector sampling point, θ r Represents the angle of the rth sampling point.

[0074] Step 4: Estimate the desired signal steering vector:

[0075] Get the preset steering vector of the desired signal And use the interference plus noise covariance matrix reconstructed in step 3 to estimate the steering vector bias:

[0076]

[0077]

[0078]

[0079] Among them, e ⊥ represents the steering vector error.

[0080] That is, based on the steering vector error e obtained by the solution ⊥ , and compare it with Add up to get the estimate of the true steering vector

[0081] Step 5: Calculate the adaptive beamforming weight vector:

[0082]

[0083] Based on the adaptive beamforming weight vector w, the array output of each snapshot can be obtained as y(n)=w(n)X(n).

[0084] To verify the performance of the method of the present invention, two sets of simulation experiments were designed under different error conditions. The number of Monte Carlo experiments was 500 in each case. The interference-to-noise ratio (INR) was set to 30 dB. The method of the present invention was compared with the LW algorithm, the Worst-Case algorithm, the diagonal loading algorithm, the LW-Rec algorithm, and the Eigenspace algorithm.

[0085] LW algorithm: L.Du, J.Li, and P.Stoica, "Fully automatic computation of diagonal loading levels for robust adaptive beamforming," IEEE Transactions on Aerospace and Electronic Systems, vol.46, no.1, pp.449–458, Jan.2010.

[0086] Worst-Case algorithm: J.Li, P.Stoica, and Z.Wang, "On robust Capon beamforming and diagonal loading," IEEE Trans.Signal Process., vol.51, no.7, pp.1702–1715, Jul.2003.

[0087] LW-Rec algorithm: Y.Gu and A.Leshem, "Robust adaptive beamforming based on jointly estimating covariance matrix and steering vector," in 2011IEEEInternational Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2011, pp.2640–2643.

[0088] Diagonal loading algorithm: BD Carlson, “Covariance matrix estimation errors and diagonal loading in adaptive arrays,” IEEE Transactions on Aerospace and Electronic systems, vol. 24, no. 4, pp. 397–401, April 1988.

[0089] Eigenspace algorithm: F.Huang, W.Sheng, and X.Ma, "Modified projection approach for robust adaptive array beamforming," Signal Processing, vol.92, no.7, pp.1758–1763, 2012.

[0090] The first set of simulation experiments was conducted under the condition of random errors in the steering vector. In the experiments verifying different input SNRs, the number of snapshots was set to 20, the SNR simulation range was -30dB to 40dB, and the changes in the array output SINR were observed. In the experiments verifying different small snapshot numbers, the SNR was set to 30dB, and the changes in the array output SINR were observed. The simulation results are shown as follows: Figure 2 and Figure 3 shown.

[0091] The second set of simulation experiments was conducted under the condition of incoherent scattering error of the steering vector. In this case, the desired signal steering vector is modeled as:

[0092]

[0093] where s p (k)(p=1,2,3,4) is a Gaussian random process independent of the random generator N(0,1).p For a Gaussian variable with distribution N(0,4), the calculated weights are changed to:

[0094]

[0095] In the experiment to verify different input SNR, the number of snapshots was set to 20, the SNR simulation range was -30dB to 40dB, and the change of array output SINR was observed. In the experiment to verify different small snapshot numbers, the SNR was set to 30dB, and the change of array output SINR was observed. The simulation results are as follows: Figure 4 and Figure 5 shown.

[0096] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the various embodiments of the present invention.

[0097] The above are only some embodiments of the present invention. For those skilled in the art, several modifications and improvements can be made without departing from the inventive concept of the present invention, which all fall within the scope of protection of the present invention.

Claims

1. A small sample adaptive beamforming method based on covariance matrix reconstruction, characterized in that: The following steps are involved: Step 1: Setting an antenna array, wherein the antenna array is a uniform linear array including M array elements, the first array element of the antenna array is a phase reference point, and the array element spacing is d=λ / 2, where λ represents the signal wavelength; Define K as the number of signal sources, where K signal sources include one desired signal and K-1 interference signals, and define the array incident angle of each signal as θ k , k=0,1,…,K-1,K>1; Define Θ s represents the angular sector of the desired signal, Θ i represents the angular sector of the interference signal, Θ n represents the noise sector, X(n) represents the array observation data vector; Step 2, Capon spectrum estimation in dimensionality reduction space: Calculate the covariance matrix of the array observation data vector X(n) in the snapshot number N Construct a random dimensionality reduction matrix based on the array structure: Among them, the random matrix Z represents a preset Gaussian random matrix, which obeys the distribution I M-1 , I L-1 Represents the M-1 and L-1 dimensional unit matrices, 0 M-1 , 0 L-1 Represent the M-1 and L-1 dimensional 0 vectors respectively, d(θ) represents the steering vector in the direction θ, I is the M-dimensional identity matrix, e1 is the unit vector with the first element being 1 and the rest being 0, L represents the preset target dimension for dimensionality reduction, and L≤min(M,N); Based on the random matrix Φ to the covariance matrix The dimension reduction is performed with the steering vector d(θ), and then the spatial power spectrum is estimated. The result of the spatial power spectrum estimation is: in, Indicates that the variable is the mean of Z; Step 3: Reconstruct the interference plus noise covariance matrix: In the noise angle sector Θ n The noise power and noise covariance matrix are calculated by performing Capon spectral integration: Noise power for: The noise covariance matrix is Among them, l(Θ n ) represents the size of the noise sector; Perform spectral integration in the interference angle sector to obtain the interference covariance matrix, and then use Add together to get the estimated noise plus interference covariance matrix Step 4: Estimate the desired signal steering vector: definition Represents the preset steering vector, e ⊥ represents the steering vector error; To maximize the output signal-to-interference-noise ratio of the beamformer, solve the quadratic constrained quadratic programming problem and obtain the steering vector error e ⊥ The value of The quadratically constrained quadratic programming problem is: The steering vector error e ⊥ With preset guide vector Add up to get the estimate of the true steering vector Step 5: Based on the obtained interference plus noise covariance matrix and estimated values Calculate adaptive beamforming weight vector Thus, the array output y(n)=w(n)X(n) for each snapshot is obtained.

2. The method according to claim 1, wherein In step 2, the covariance matrix of the array observation data vector X(n) in the snapshot number N is for:

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