DOA estimation method for coprime arrays based on information geometry

Abstract

The present invention belongs to the field of radar system direction of arrival estimation, and aims to obtain higher estimation accuracy under the condition of the same number of array elements. To this end, the technical solution adopted by the present invention is a coprime array DOA estimation method based on information geometry, which uses the covariance matrix of the virtual array received signal as the parameter space on the basis of the probability distribution geometry structure, and estimates the arrival angle by minimizing the difference between the theoretical value and the actual value of the probability density function of the received signal. The present invention is mainly used in radar system direction of arrival estimation occasions.

Description

Technical Field

[0001] The invention relates to the field of radar system direction of arrival estimation, and in particular to a coprime array DOA estimation method based on information geometry. Background Art

[0002] A basic problem in array signal processing is the estimation of the direction of arrival (DOA) of spatial signals, which is also one of the important tasks in many fields such as radar and sonar. Early methods for DOA estimation also include the maximum likelihood spectrum estimation method based on statistical analysis, the maximum entropy spectrum estimation method, and the autoregression moving average (ARMA) spectrum analysis method, but the computational complexity of these methods is generally high and does not fully match the requirements of practical applications. In 1979, Schmidt et al. [1] proposed the Multiple Signal Classification (MUSIC) algorithm, which uses the orthogonality of the signal subspace and the noise subspace to divide the array covariance matrix into two spaces for parameter estimation. This is a milestone in the field of array signal processing. It breaks through the Rayleigh limit of traditional methods and achieves a qualitative leap in the performance of direction of arrival estimation. In 1986, Roy et al. [2] proposed the Estimating Signal Parameter via Rotational Invariance Techniques (ESPRIT) algorithm. The basic idea is to study the rotational invariance of the signal subspace caused by the shift-invariant characteristics of the array. This algorithm does not require spectral peak search and has lower computational complexity than the MUSIC algorithm. In 1997, Gorodnitsky et al. [3] proposed the Focal Underdetermined System Estimation (Focal Underdetermined System Estimation) algorithm. Solver, FOCUSS), which uses the sparsity of the signal to perform statistical signal processing, avoiding the performance degradation caused by subspace perturbations in low signal-to-noise ratio conditions, and can solve coherent signals with strong robustness.

[0003] Direction of arrival estimation has been extensively studied based on the above-mentioned landmark representative algorithms, and a series of optimized algorithm designs have been formed from different perspectives. It is noted that in the existing direction of arrival estimation algorithms, the uniform linear array is the most commonly used array structure. However, when using a uniform linear array for direction of arrival estimation, its degree of freedom performance is limited by the number of physical array elements. At the same time, a larger array aperture is the guarantee for achieving the resolution and accuracy of direction of arrival estimation. In order to obtain higher degrees of freedom and better direction of arrival estimation effects, traditional algorithms need to be implemented by adding physical array elements, resulting in increased hardware complexity and computational complexity. Therefore, there is a certain trade-off between the algorithm performance and complexity of direction of arrival estimation.

[0004] Coprime array is one of the typical sparse array structures in the spatial domain under the framework of coprime sampling technology. The sparse array architecture can be realized by selecting a pair of coprime integers [4]. Compared with the traditional uniform linear array, the array elements in the coprime array are sparsely placed, which greatly increases the array aperture, reduces the mutual coupling effect between the array elements, and improves the estimation accuracy and resolution. In addition, we can derive a set of virtual uniform linear arrays containing more virtual array elements based on the properties of prime numbers, thereby effectively breaking through the limitation of the number of physical array elements on the degree of freedom in the traditional uniform linear array structure. Therefore, when using the same number of physical array elements, the coprime array can obtain higher resolution and greater degrees of freedom than the uniform linear array.

[0005] Information geometry (IG) is a theoretical system that studies the intrinsic geometric properties of statistical manifolds composed of probability distribution families. It regards the concepts in probability theory and statistics as the geometric structure of statistical manifolds, and thus uses differential geometry methods to study the structural properties on manifolds. STJensen et al. [5] proposed a method to represent Fisher information distance. Based on this method, Mario et al. [6] proposed a new DOA estimation method based on information geometry, the IG method, which can provide higher resolution than traditional methods. The IG method takes into account the geometric shape of the probability distribution space after the covariance matrix is ​​parameterized, uses the geodesic distance criterion to establish an optimization problem, and solves it using the linear search method. However, through analysis, it is found that the performance of the IG method is related to the number of array elements, signal power, and noise power. Only when certain conditions are met can the estimation be accurate.

[0006] [1] R.Schmidt, "Multiple emitter location and signal parameter estimation," in IEEE Transactions on Antennas and Propagation, vol.34, no.3, pp.276-280, March 1986, doi:10.1109 / TAP.1986.1143830.

[0007] [2] R.Roy and T.Kailath, "ESPRIT-estimation of signal parameters viarotational invariance techniques," in IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.37, no.7, pp.984-995, July 1989, doi:10.1109 / 29.32276.

[0008] [3]IFGorodnitsky and BDRao, "Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm," in IEEETransactions on Signal Processing, vol.45, no.3, pp.600-616, March 1997, doi:10.1109 / 78.558475.

[0009] [4] Shen Yifan. Research on gridless direction of arrival estimation algorithm based on coprime array[D]. Zhejiang University, 2017.

[0010] [5]STJensen,private communication in[2],1976.

[0011] [6] M. Coutino, R. Pribic and G. Leus, "Direction of arrival estimation based on information geometry," 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2016, pp.3066-3070, doi:10.1109 / ICASSP.2016.7472241. Summary of the invention

[0012] In order to overcome the shortcomings of the prior art, the present invention aims to remove the restrictions on the number of array elements, signal power and noise power by normalizing the covariance matrix of the actual received signal, and then apply this method to the covariance matrix of the virtual array based on the coprime array, so as to obtain a higher estimation accuracy under the condition of the same number of array elements. To this end, the technical solution adopted by the present invention is a coprime array DOA estimation method based on information geometry, which uses the covariance matrix of the virtual array received signal as the parameter space on the basis of the probability distribution geometry structure, and estimates the arrival angle by minimizing the difference between the theoretical value and the actual value of the probability density function of the received signal.

[0013] The specific steps are as follows:

[0014] Step 1: Solve the actual covariance matrix of the received signal of the coprime array composed of (2M+N-1) array elements

[0015] Step 2: Construct a virtual array using coprime arrays

[0016] Step 3: Using the actual covariance matrix Solve the covariance matrix R of the received signal under the virtual array xx ;

[0017] Step 4: By the covariance matrix R xx Perform singular value decomposition to obtain the transmitted signal power P t And the noise power

[0018] Step 5: Covariance matrix R xx After normalization, we get

[0019] Step 6: Construct the covariance matrix of the virtual array subarray corresponding to the virtual array element positions from 0 to MNd

[0020] Step 7: Solve to obtain Φ such that and The Fisher information distance between them is the smallest.

[0021] Step 1: Array 1 is composed of 2M array antennas with a spacing of Nd, and array 2 is composed of N array antennas with a spacing of Md, where M and N are coprime integers and d is half a wavelength. These two sparse arrays are combined to obtain a coprime array configuration of (2M+N-1);

[0022] The actual covariance matrix obtained from the coprime array received signal is:

[0023]

[0024] where K is the number of snapshots, y[k] is the received signal matrix, and [·] H represents the conjugate transpose of a matrix;

[0025] Step 2: The theoretical covariance matrix constructed based on the coprime array received signal is:

[0026]

[0027] in, P is the power of the signal source, is the noise signal power, I is the unit matrix;

[0028] The covariance matrix of the coprime array received signal is vectorized to get

[0029]

[0030] Among them, z corresponds to a virtual array, and the position of each element in the virtual array is

[0031] S(x,y)={u x -u y |x,y=1,2,…,2M+N-1}

[0032] The values ​​of S(x,y) are repeated. After removing the overlapping virtual array elements at each position, a non-uniform virtual array S is obtained. c , its equivalent virtual signal can be obtained by selecting the elements at the corresponding positions in the vector z; select this section of non-uniform virtual array S c A continuous uniform virtual array element with 0 as the center and the longest length forms a uniform virtual array containing (2MN+1) virtual array elements. The actual covariance matrix Instead of the ideal covariance matrix R y Construct a virtual array;

[0033] Step 3: The corresponding virtual linear array Y from (MN)d to -(MN)d is divided into several partially overlapping sub-arrays V i .V i Corresponding to the virtual array elements at positions (-i+1)d to (-i+MN+1)d in Y, the equivalent virtual received signal vector of each virtual subarray is constructed by selecting elements in rows (MN+2-i) to (2MN+2-i) in vector z, expressed as Virtual array covariance matrix R based on the original coprime array xx , which can be obtained by taking the arithmetic square root of the single-rank spatial smoothing covariance matrix of all virtual sub-matrices:

[0034]

[0035] Step 4: R xx Perform eigenvalue decomposition to obtain the transmission power and noise power

[0036]

[0037]

[0038] in, Represents the eigendecomposition process, and the eigenvalues ​​are in the order of λ1≥λ2=…=λ M Sort, is the corresponding eigenvector;

[0039] Step 5: Covariance matrix R xx After normalization, we get

[0040] Step 6: Construct the covariance matrix of the virtual array subarray corresponding to the virtual array element positions from 0 to MNd

[0041]

[0042] in, because After normalization, R(Φ) should also be written in the following form

[0043] Step 7: Use the method of minimizing R(Φ) and The Fisher information distance between them is solved to obtain Φ

[0044]

[0045] in, a1,…,a n yes When the rank of R(Φ) is 1, the formula can be simplified to the following form

[0046]

[0047] Finally, the coprime array DOA estimation method based on information geometry becomes the following optimization problem:

[0048]

[0049] Then, the DOA is estimated by using spectrum peak search.

[0050] The characteristics and beneficial effects of the present invention are:

[0051] The coprime array DOA estimation method based on information geometry proposed in the present invention removes the restrictions of the IG method on the number of array elements, signal power and noise power. In the simulation experiment, the number of array elements is set to 8 and the signal-to-noise ratio is set to 20dB. Figure 4 The comparison of DOA estimation accuracy between this algorithm and the IG method at different transmit powers is shown. It can be seen that the estimation accuracy of the IG method is better than that of this algorithm only when the transmit power is 1. As the transmit power increases, the difference in estimation accuracy between the algorithms also increases. Description of the drawings:

[0052] Figure 1 Mutually prime array formation process.

[0053] Figure 2 Schematic diagram of spatial smoothing technique.

[0054] Figure 3 Algorithm steps.

[0055] Figure 4 Comparison of DOA estimation accuracy between this algorithm and the IG algorithm at different transmit powers. DETAILED DESCRIPTION

[0056] The invention belongs to the field of radar system direction of arrival estimation, designs a coprime array DOA estimation method based on information geometry, and uses the Fisher information distance in information geometry to measure the difference between probability density functions of received signals.

[0057] The performance of the IG method proposed by Mario et al. is related to the number of array elements, signal power and noise power. In order to overcome the shortcomings of the prior art, the present invention normalizes the covariance matrix of the actual received signal to remove the restrictions of the IG method on the number of array elements, signal power and noise power, and then applies this method to the virtual array covariance matrix based on coprime arrays, thereby obtaining higher estimation accuracy under the same number of array elements.

[0058] The present invention aims to propose a coprime array DOA estimation method based on information geometry. Based on the probability distribution geometry structure, the covariance matrix of the virtual array received signal is used as the parameter space to estimate the arrival angle by minimizing the difference between the theoretical value and the actual value of the probability density function of the received signal. The specific steps of the technical solution adopted by the present invention are as follows:

[0059] Step 1: Solve the actual covariance matrix of the received signal of the coprime array composed of (2M+N-1) array elements

[0060] Step 2: Construct a virtual array using coprime arrays

[0061] Step 3: Using the actual covariance matrix Solve the covariance matrix R of the received signal under the virtual array xx ;

[0062] Step 4: By the covariance matrix R xx Perform singular value decomposition to obtain the transmitted signal power P t And the noise power

[0063] Step 5: Covariance matrix R xx After normalization, we get

[0064] Step 6: Construct the covariance matrix of the virtual array subarray corresponding to the virtual array element positions from 0 to MNd

[0065] Step 7: Solve to obtain Φ such that and The Fisher information distance between them is the smallest;

[0066] The present invention is further described in detail below with reference to the accompanying drawings and specific examples.

[0067] Take estimating a source as an example:

[0068] Step 1: Array 1 consists of 2M array antennas with a spacing of Nd, and array 2 consists of N array antennas with a spacing of Md, such as Figure 1 As shown. M and N are coprime integers, and d is half wavelength. Combining these two sparse arrays, we get a coprime array configuration of (2M+N-1).

[0069] The actual covariance matrix obtained from the coprime array received signal is:

[0070]

[0071] where K is the number of snapshots, y[k] is the received signal matrix, and [·] H represents the conjugate transpose of a matrix;

[0072] Step 2: The theoretical covariance matrix constructed based on the coprime array received signal is:

[0073]

[0074] in, P is the power of the signal source, is the noise signal power and I is the identity matrix.

[0075] The covariance matrix of the coprime array received signal is vectorized to get

[0076]

[0077] Among them, z corresponds to a virtual array, and the position of each element in the virtual array is

[0078] S(x,y)={u x -u y |x,y=1,2,…,2M+N-1}

[0079] The values ​​of S(x,y) are repeated. After removing the overlapping virtual array elements at each position, a non-uniform virtual array S is obtained. c , its equivalent virtual signal can be obtained by selecting the elements at the corresponding positions in the vector z. Select this section of non-uniform virtual array S c A continuous uniform virtual array element with 0 as the center and the longest length forms a uniform virtual array containing (2MN+1) virtual array elements. The actual covariance matrix Instead of the ideal covariance matrix R y Construct a virtual array.

[0080] Step 3: The corresponding virtual linear array Y from (MN)d to -(MN)d is divided into several partially overlapping sub-arrays V i .V i Corresponding to the virtual array elements at positions (-i+1)d to (-i+MN+1)d in Y, the equivalent virtual received signal vector of each virtual subarray is constructed by selecting elements in rows (MN+2-i) to (2MN+2-i) in vector z, expressed as Virtual array covariance matrix R based on the original coprime array xx , which can be obtained by taking the arithmetic square root of the single-rank spatial smoothing covariance matrix of all virtual sub-matrices:

[0081]

[0082] Step 4: R xx Perform eigenvalue decomposition to obtain the transmission power and noise power

[0083]

[0084]

[0085] in, Represents the eigendecomposition process, and the eigenvalues ​​are in the order of λ1≥λ2=…=λ M Sorting, is the corresponding eigenvector.

[0086] Step 5: Covariance matrix R xx After normalization, we get

[0087] Step 6: Construct the covariance matrix of the virtual array subarray corresponding to the virtual array element positions from 0 to MNd

[0088]

[0089] in, because After normalization, R(Φ) should also be written in the following form

[0090] Step 7: Use the method of minimizing R(Φ) and The Fisher information distance between them is solved to obtain Φ.

[0091]

[0092] in, a1,…,a n yes When the rank of R(Φ) is 1, the formula can be simplified to the following form

[0093]

[0094] Finally, the coprime array DOA estimation method based on information geometry becomes the following optimization problem:

[0095]

[0096] Then, the DOA is estimated by using spectrum peak search.

[0097] The above description is only a specific implementation mode of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by any technician familiar with the technical field within the technical scope disclosed in the present invention should be covered within the protection scope of the present invention.

Claims

1. A coprime array DOA estimation method based on information geometry, characterized in that Based on the geometric structure of probability distribution, the covariance matrix of the virtual array received signal is used as the parameter space to estimate the arrival angle by minimizing the difference between the theoretical value and the actual value of the probability density function of the received signal. The specific steps are as follows: Step 1: Solve the actual covariance matrix of the received signal of the coprime array composed of (2M+N-1) array elements Step 2: Construct a virtual array using coprime arrays Step 3: Using the actual covariance matrix Solve the covariance matrix R of the received signal under the virtual array xx ; Step 4: By the covariance matrix R xx Perform singular value decomposition to obtain the transmitted signal power P t And the noise power Step 5: Covariance matrix R xx After normalization, we get I is the identity matrix; Step 6: Construct the covariance matrix of the virtual array subarray corresponding to the virtual array element positions from 0 to MNd Step 7: Solve to obtain Φ such that and The Fisher information distance between them is the smallest.

2. The method for estimating DOA of coprime arrays based on information geometry as claimed in claim 1, wherein the steps 1 : Array 1 is composed of 2M array antennas with a spacing of Nd, and array 2 is composed of N array antennas with a spacing of Md, where M and N are coprime integers and d is a half wavelength. Combining these two sparse arrays, we get a coprime array configuration of (2M+N-1); The actual covariance matrix obtained from the coprime array received signal is: where K is the number of snapshots, y[k] is the received signal matrix, and [·] H represents the conjugate transpose of a matrix; Step 2: The theoretical covariance matrix constructed based on the coprime array received signal is: in, P is the power of the signal source; The covariance matrix of the coprime array received signal is vectorized to get Among them, z corresponds to a virtual array, and the position of each element in the virtual array is S(x,y)={u x -u y ∣x,y=1,2,…,2M+N-1} The values ​​of S(x,y) are repeated. After removing the overlapping virtual array elements at each position, a non-uniform virtual array S is obtained. c , its equivalent virtual signal can be obtained by selecting the elements at the corresponding positions in the vector z; select this section of non-uniform virtual array S c A continuous uniform virtual array element with 0 as the center and the longest length forms a uniform virtual array containing (2MN+1) virtual array elements. The actual covariance matrix Instead of the ideal covariance matrix R y Construct a virtual array; Step 3: The corresponding virtual linear array Y from (MN)d to -(MN)d is divided into several partially overlapping sub-arrays V i .V i Corresponding to the virtual array elements at positions (-i+1)d to (-i+MN+1)d in Y, the equivalent virtual received signal vector of each virtual subarray is constructed by selecting elements in rows (MN+2-i) to (2MN+2-i) in vector z, expressed as Virtual array covariance matrix R based on the original coprime array xx , which can be obtained by taking the arithmetic square root of the single-rank spatial smoothing covariance matrix of all virtual sub-matrices: Step 4: R xx Perform eigenvalue decomposition to obtain the transmission power and noise power in, Represents the eigendecomposition process, and the eigenvalues ​​are in the order of λ1≥λ2=...=λ M Sort, is the corresponding eigenvector; Step 5: Covariance matrix R xx After normalization, we get Step 6: Construct the covariance matrix of the virtual array subarray corresponding to the virtual array element positions from 0 to MNd Where a(Φ)=[1,e -jπsin(θ) ,...,e -jπMNsin(θ) ] T ,because After normalization, R(Φ) should also be written in the following form Step 7: Use the method of minimizing R(Φ) and The Fisher information distance between them is solved to obtain Φ in, a1,…,a n yes When the rank of R(Φ) is 1, the formula can be simplified to the following form Finally, the coprime array DOA estimation method based on information geometry becomes the following optimization problem: Then, the DOA is estimated by using spectrum peak search.

Images