Low-complexity preprocessing method and direction-of-arrival estimation method based on sparse array

Abstract

The invention belongs to the field of array antenna signal processing, and particularly relates to a low-complexity dimensionality reduction preprocessing method and a direction of arrival estimation method based on a sparse array, which are characterized in that a bijection transformation matrix is constructed through a mapping relation between a physical array and a virtual array; according to the method, the virtual array single-block beat data is directly constructed by constructing the bijection transformation matrix of the sparse physical array and the virtual array, so that the problem of resource waste caused by data redundancy in a covariance matrix in a traditional method is solved, and the method has the advantages of being simple in structure, convenient to operate and high in practicability. And technical support is provided for engineering implementation of the direction of arrival.

Description

Technical field:

[0001] The invention belongs to the field of array antenna signal processing, and in particular to a low-complexity dimension reduction preprocessing method and a direction of arrival estimation method based on a sparse array. Background technology:

[0002] Direction of arrival estimation focuses on obtaining the directional information of the incoming wave to provide a basis for positioning, tracking and identification, and is widely used in communications, radar, sonar and other fields. Traditional spatial spectrum estimation methods, such as multiple signal classification methods and rotationally invariant subspace methods, have attracted much attention in research due to their excellent super-resolution angle measurement capabilities. With the increase in radiation source equipment and the increasingly complex electromagnetic environment, traditional spatial spectrum estimation methods can no longer adapt to scenarios where the number of radiation sources far exceeds the number of array elements. However, the proposed underdetermined direction of arrival estimation method based on sparse arrays effectively solves the dilemma of traditional subspace methods.

[0003] The sparse array uses the equivalent virtual array characteristics of the covariance matrix to transform the subspace into the common array space, and combines the virtual array space smoothing technology to realize the underdetermined wave arrival estimation, which has the advantages of low cost, easy array, high degree of freedom, low coupling, and high precision. However, the spatial smoothing technology can only use the uniform linear array part in the virtual array, and the non-uniform part cannot be used, resulting in a waste of computing resources. In addition, the uniform virtual array equivalent received data in this method has a serious data redundancy problem, which increases the computational complexity and causes a waste of computing resources. Therefore, it is of great research significance to eliminate the calculation process of invalid data, reduce the redundancy of valid data, and improve computing efficiency. Summary of the invention:

[0004] Aiming at the classic sparse array underdetermined direction of arrival estimation method, which needs to calculate the covariance matrix of received data and vectorize it to obtain the single snapshot received data of the equivalent virtual array, the present invention proposes a method that can directly calculate the equivalent single snapshot data of the uniform virtual array through the physical array received data, thereby avoiding the data redundancy problem caused by calculating the complete covariance matrix. A low-complexity preprocessing method based on sparse arrays and a direction of arrival estimation method are provided.

[0005] The present invention is achieved by the following measures:

[0006] A low-complexity preprocessing method based on sparse arrays is characterized in that a bijective transformation matrix is ​​constructed through the mapping relationship between the physical array and the virtual array, and the bijective transformation characteristics are used to directly obtain equivalent virtual data of the uniform part in the virtual array.

[0007] The present invention further specifically comprises the following steps:

[0008] Step 1: Construct a mapping matrix using the surjective non-injective relationship between the physical array and the virtual uniform line array, which specifically includes the following steps:

[0009] Step 1-1: Assume that the receiving physical array is a sparse linear array consisting of L array elements, and its array element positions are Where d represents the normalized array element spacing, satisfying d = λ / 2, λ is the wavelength of the received signal; Step 2: Assume that the virtual array corresponding to the sparse linear array is The number of array elements is f S , physical array The corresponding virtual array Satisfaction relationship:

[0010]

[0011] Steps 1-3: Assuming a virtual array The uniform linear array part is The number of array elements is f U , satisfying f U =2U+1

[0012] Where U is the maximum value of the array element in the continuous virtual array, and the uniform linear array and virtual arrays satisfy Step 1-4: Construct the mapping matrix I(s) using the surjective non-injective relationship between the relative positions of the physical array elements and the virtual array element positions.

[0013]

[0014] Among them, <·> i,j Represents the i-th row and j-th column element of the matrix;

[0015] Step 2: Design the bijective transformation matrix between the physical array and the virtual array;

[0016] Step 3: Receive the radiation source signal using the antenna array;

[0017] Step 4: Use the bijective transformation matrix to directly construct the single-block data;

[0018] Step 5: Perform Toeplitz reconstruction on the single-block data.

[0019] Step 2 of the present invention specifically comprises the following steps:

[0020] Step 2-1: Construct a binary bijective transformation matrix set Q(s) for each binary mapping matrix I(s) corresponding to the virtual array element position s, satisfying

[0021]

[0022] Among them, <·> i,j represents the element in the i-th row and j-th column of the matrix, represents the Frobenius norm, I(s)≥I′(s) means that the elements in the mapping matrix I(s) are not less than the elements in the same position in I′(s);

[0023] Step 2-2: Take any bijective transformation matrix I′(s) from the set Q(s) that satisfies

[0024] The single-block received data at the virtual array element position s is determined according to the i-th row and j-th column where the only non-zero element s in I′(s) is located.

[0025] Step 3 of the present invention is specifically as follows: Consider that there are Q far-field narrowband signals incident on the array in space, where it is assumed that Q is known a priori, d satisfies d≤λ / 2 to avoid phase ambiguity, λ is the wavelength of the narrowband signal, and the array receiving signal is:

[0026] Among them, A(θ) is the L×Q-dimensional array flow matrix, S(t) is the Q×1-dimensional incident signal vector, N(t) is the L×1-dimensional additive Gaussian white noise vector, and a(θ) is the column vector of A(θ), which can be expressed as:

[0027]

[0028] Step 4 of the present invention specifically comprises the following steps:

[0029] Step 4-1: Using the positional relationship between the i-th row and j-th column of the only non-zero element in the bijective transformation matrix I′(s) and the virtual array element s, the received data y at the virtual array element s is directly calculated using the received data of the i-th antenna and the j-th antenna. s (t), satisfying

[0030]

[0031] in,(·) H represents the conjugate transpose, is the additional noise, p i -p j is the corresponding delay s;

[0032] Step 4-2: Calculate f U The virtual array elements receive data and arrange them in descending order according to s∈[-U,U]y s (t), the single-block received data Y(t) under the virtual array satisfies

[0033]

[0034] Among them, B=[b(θ1),b(θ2),…,b(θ Q )] represents the array flow type of the virtual array, represents the single snapshot signal received under the equivalent virtual array, which is composed of the power of the actual signal; represents the steering vector of the virtual array; represents the noise added to the virtual array, which can be expressed as

[0035] Step 5 of the present invention is specifically: dividing the array obtained in step 4 into U+1 overlapping subarrays, each subarray has U+1 elements, and the position of the kth element in the i-th subarray in the original array is:

[0036] {(i+kU-2)d|i,k=1,...,U+1}, the received data Y of the i-th subarray i (t) is Y i (t) = [y i-U-1 (t),…,y i-1 (t)] T , Toeplitz reconstruction is performed using the received data of U+1 sub-arrays to construct the covariance matrix R Y Satisfy R Y =[Y1(t),Y2(t),…,Y U+1 (t)].

[0037] The present invention also proposes a method for estimating the direction of arrival based on a low-complexity preprocessing method of a sparse matrix, which is characterized in that a signal subspace or a noise subspace is obtained according to the number of information sources to obtain the signal direction of arrival, and specifically comprises the following steps:

[0038] Step 6-1: Restore the covariance matrix of the rank Perform eigenvalue decomposition

[0039] Among them, U represents the feature space, Σ represents the eigenvalue, and U s represents the signal space, U n represents the noise subspace, Σ s represents the eigenvalue corresponding to the signal subspace, Σ n Represents the eigenvalue corresponding to the noise subspace;

[0040] Step 6-2: Using the orthogonality of the two subspaces, construct a search function to perform angle search.

[0041] f MUSIC The θ corresponding to the Q maximum values ​​of (θ) is the arrival direction.

[0042] Compared with the prior art, the present invention utilizes the mapping relationship between the physical sparse array and the virtual uniform array, constructs a bijective matrix between the two, and directly calculates the equivalent single snapshot data of the virtual uniform array, thereby solving the problem of data redundancy in preprocessing, effectively improving the calculation efficiency, and providing a theoretical reference for the practical engineering of the underdetermined direction finding algorithm based on sparse array. Description of the drawings:

[0043] Figure 1 It is a flow chart of the present invention.

[0044] Figure 2 This is the physical array structure of a 30-element k-order extended coprime array adopted in an embodiment of the present invention, wherein k=8, M=2, and N=7.

[0045] Figure 3 Schematic diagram of a virtual array of a k-order extended coprime array with 31 elements.

[0046] Figure 4 A schematic diagram of the positive half axis of the virtual array structure calculated by the preprocessing method proposed in the present invention.

[0047] Figure 5 In the embodiment of the present invention, 31 far-field narrowband uncorrelated signals are incident on Figure 1 As shown in the figure, when the k-order extended coprime array with 30 array elements is used, the RMSE of the method proposed in the present invention and the traditional method changes with the input SNR, wherein the signal-to-noise ratio changes from -20dB to 30dB in steps of 5dB, the number of snapshots K=10000, and the incident angle is [-60°:5°:60°]. Specific implementation method:

[0048] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0049] This example proposes a fast preprocessing algorithm for dimensionality reduction based on second-order difference common array and an underdetermined direction of arrival estimation method for far-field narrowband radio signals. The specific steps are as follows: Figure 1 As shown, including:

[0050] The first step is to construct a mapping matrix using the surjective non-injective relationship between the physical array and the virtual uniform line array. The first step includes the following steps:

[0051] Step 1-1: Assume that the receiving physical array is a sparse linear array consisting of L array elements, and its array element positions are Wherein, d represents the normalized array element spacing, satisfying d = λ / 2, λ is the wavelength of the received signal;

[0052] Step 1-2: Assume that the virtual array corresponding to the sparse linear array is The number of array elements is f S . Physical Array The corresponding virtual array Satisfaction relationship,

[0053]

[0054] Steps 1-3: Assuming a virtual array The uniform linear array part is The number of array elements is f U , satisfying f U =2U+1, where U is the value of the maximum element in the continuous virtual array. and virtual arrays satisfy, Step 1-4: Construct the mapping matrix I(s) using the surjective non-injective relationship between the relative positions of the physical array elements and the virtual array element positions.

[0055] Among them, <·> i,j Represents the i-th row and j-th column element of the matrix;

[0056] The second step is to design the bijective transformation matrix of the physical array and the virtual array. The second step includes the following steps: Step 2-1: Construct a binary bijective transformation matrix set Q(s) for each binary mapping matrix I(s) corresponding to the virtual array element position s, satisfying,

[0057]

[0058] Among them, <·> i,j represents the element in the i-th row and j-th column of the matrix, represents the Frobenius norm, I(s)≥I′(s) means that the elements in the mapping matrix I(s) are not less than the elements in the same position in I′(s);

[0059] Step 2-2: Take any bijective transformation matrix I′(s) in the set Q(s) that satisfies: Determine the single-block received data at the virtual array element position s according to the i-th row and j-th column where the only non-zero element s in I′(s) is located;

[0060] The third step is to use the antenna array to receive the radiation source signal, which specifically includes the following steps: Consider that there are Q far-field narrowband signals in space incident on the array, where Q is assumed to be known a priori, d satisfies d≤λ / 2 to avoid phase ambiguity, and λ is the wavelength of the narrowband signal. The array receives the signal:

[0061]

[0062] Among them, A(θ) is the L×Q-dimensional array flow matrix, S(t) is the Q×1-dimensional incident signal vector, N(t) is the L×1-dimensional additive Gaussian white noise vector, and a(θ) is the column vector of A(θ), which can be expressed as:

[0063]

[0064] The fourth step is to directly construct single-block data using a bijective transformation matrix. The fourth step includes the following steps:

[0065] Step 4-1: Using the positional relationship between the i-th row and j-th column of the only non-zero element in the bijective transformation matrix I′(s) and the virtual array element s, the received data y at the virtual array element s is directly calculated using the received data of the i-th antenna and the j-th antenna. s (t), satisfied,

[0066]

[0067] in,(·) H represents the conjugate transpose, is the additional noise, p i -p j is the corresponding delay s;

[0068] Step 4-2: Calculate f U The virtual array elements receive data and arrange them in descending order according to s∈[-U,U]y s (t), the single-block received data Y(t) under the virtual array satisfies,

[0069]

[0070] Among them, B=[b(θ1),b(θ2),…,b(θ Q )] represents the array flow type of the virtual array, represents the single snapshot signal received under the equivalent virtual array, which is composed of the power of the actual signal; represents the steering vector of the virtual array; represents the noise added to the virtual array, expressed as,

[0071]

[0072] The fifth step is to perform Toeplitz reconstruction on the single-block data. The fifth step includes the following steps: the array is divided into U+1 overlapping subarrays, each subarray has U+1 elements, and the position of the kth element in the i-th subarray in the original array is:

[0073] {(i+kU-2)d|i,k=1,...,U+1},

[0074] The received data Y of the i-th subarray i (t) is Y i (t) = [y i-U-1 (t),…,y i-1 (t)] T ,

[0075] The received data of U+1 sub-arrays are used for Toeplitz reconstruction to construct the covariance matrix R Y Satisfied, R Y =[Y1(t),Y2(t),…,Y U+1 (t)];

[0076] The sixth step is to obtain a signal subspace or a noise subspace according to the number of information sources, and obtain a signal arrival direction. The sixth step includes the following steps:

[0077] The covariance matrix of the restored rank Perform eigenvalue decomposition,

[0078]

[0079] Among them, U represents the feature space, Σ represents the eigenvalue, and U s represents the signal space, U n represents the noise subspace, Σ s represents the eigenvalue corresponding to the signal subspace, Σ n Represents the eigenvalue corresponding to the noise subspace; using the orthogonality of the two subspaces, a search function is constructed to perform angle search.

[0080] f MUSIC The θ corresponding to the Q maximum values ​​of (θ) is the arrival direction.

[0081] The performance of the present invention can be illustrated by the following analysis, comparison and simulation:

[0082] 1. Computational complexity analysis: To facilitate the comparative analysis of computational complexity and highlight the advantages of the computational complexity of the present invention, the classical virtual array wave arrival estimation direction is selected for comparison. Among them, the traditional method includes six steps: calculation of the complete covariance matrix, vectorization, equivalent single snapshot data calculation, spatial smoothing, eigenvalue decomposition, and spectrum search. The computational complexity required by the traditional method is O(f S 2 K+f U f S 2 +U 3 +U 3 +U 2 J), where f S Indicates the number of physical array elements, f Urepresents the number of array elements of the virtual uniform array, U is the maximum value of the virtual uniform array, and J represents the number of spectrum search points.

[0083] The method proposed in the present invention includes three steps: directly calculating equivalent single snapshot data using bijective matrix, matrix reconstruction, eigenvalue decomposition and spatial spectrum search. The required computational complexity is:

[0084] O(f U K+U 3 +U 2 J),

[0085] Among them, f U represents the number of array elements of the virtual uniform array, U is the maximum value of the virtual uniform array, and J represents the number of spectrum search points.

[0086] Table 1 Computational complexity analysis

[0087]

[0088] 2. Simulation conditions

[0089] Assume that 31 far-field narrowband uncorrelated signals are incident on a k-times Extended Coprime Array (k-ECA) structure consisting of 30 array elements at angles uniformly distributed between [-60°, 60°]. The k-times Extended Coprime Array consists of two uniform linear arrays, and the physical array structure is:

[0090]

[0091] Wherein, M and N are a pair of mutually prime integers, respectively representing the element spacing of the uniform line subarray. Integers m and n satisfy m∈[0,kM-1], n∈[0,N-1], representing the specific positions of the subarray elements. Wherein, the integer k represents the expansion multiple. In order to further evaluate the performance of the present invention, the number of Monte Carlo experiments is set to 500, and the root mean square error (RMSE) is used as the evaluation index.

[0092] 3. Simulation content and results:

[0093] Simulation 1, a sparse array of a k-order extended coprime array with 30 elements, with parameters k=8, M=2, N=7, and the physical array structure is as follows Figure 2 As shown; the equivalent single snapshot data calculated using the traditional method, the positive half axis schematic diagram of the virtual array structure is as follows Figure 3 The equivalent single snapshot data calculated by the preprocessing method proposed in the present invention has a positive semi-axis schematic diagram of the virtual array structure as shown in FIG. Figure 4 shown.

Claims

1. A low-complexity preprocessing method based on sparse array, characterized in that: Through the mapping relationship between the physical array and the virtual array, a bijective transformation matrix is ​​constructed, and the bijective transformation characteristics are used to directly obtain the equivalent virtual data of the uniform part in the virtual array.

2. The low-complexity preprocessing method based on sparse array according to claim 1, characterized in that: The specific steps include: Step 1: Construct a mapping matrix using the surjective non-injective relationship between the physical array and the virtual uniform line array, which specifically includes the following steps: Step 1-1: Assume that the receiving physical array is a sparse linear array consisting of L array elements, and its array element positions are Wherein, d represents the normalized array element spacing, satisfying d = λ / 2, λ is the wavelength of the received signal; Step 2: Assume that the virtual array corresponding to the sparse linear array is The number of array elements is f S , physical array The corresponding virtual array Satisfaction relationship: Steps 1-3: Assuming a virtual array The uniform linear array part is The number of array elements is f U , satisfying f U =2U+1 Where U is the maximum value of the array element in the continuous virtual array, and the uniform linear array and virtual arrays satisfy Step 1-4: Construct the mapping matrix I(s) using the surjective non-injective relationship between the relative positions of the physical array elements and the virtual array element positions. Among them, <·> i,j Represents the i-th row and j-th column element of the matrix; Step 2: Design the bijective transformation matrix between the physical array and the virtual array; Step 3: Receive the radiation source signal using the antenna array; Step 4: Use the bijective transformation matrix to directly construct the single-block data; Step 5: Perform Toeplitz reconstruction on the single-block data.

3. The low-complexity preprocessing method based on sparse array according to claim 2, characterized in that: Step 2 specifically includes the following steps: Step 2-1: Construct a binary bijective transformation matrix set Q(s) for each binary mapping matrix I(s) corresponding to the virtual array element position s, satisfying Among them, <·> i,j represents the element in the i-th row and j-th column of the matrix, represents the Frobenius norm, I(s)≥I′(s) means that the elements in the mapping matrix I(s) are not less than the elements in the same position in I′(s); Step 2-2: Take any bijective transformation matrix I′(s) from the set Q(s) that satisfies The single-block received data at the virtual array element position s is determined according to the i-th row and j-th column where the only non-zero element s in I′(s) is located.

4. The low-complexity preprocessing method based on sparse array according to claim 2, characterized in that: In step 3, consider that there are Q far-field narrowband signals incident on the array in space, where Q is assumed to be known a priori, d satisfies d≤λ / 2 to avoid phase ambiguity, λ is the wavelength of the narrowband signal, and the array receives the signal: Among them, A(θ) is the L×Q-dimensional array flow matrix, S(t) is the Q×1-dimensional incident signal vector, N(t) is the L×1-dimensional additive Gaussian white noise vector, and a(θ) is the column vector of A(θ), which can be expressed as:

5. The low-complexity preprocessing method based on sparse array according to claim 2, characterized in that: Step 4 specifically includes the following steps: Step 4-1: Using the positional relationship between the i-th row and j-th column of the only non-zero element in the bijective transformation matrix I′(s) and the virtual array element s, the received data y at the virtual array element s is directly calculated using the received data of the i-th antenna and the j-th antenna. s (t), satisfying in,(·) H represents the conjugate transpose, is the additional noise, p i -p j is the corresponding delay s; Step 4-2: Calculate f U The virtual array elements receive data and arrange them in descending order according to s∈[-U,U]y s (t), the single-block received data Y(t) under the virtual array satisfies Among them, B=[b(θ1),b(θ2),…,b(θ Q )] represents the array flow type of the virtual array, represents the single snapshot signal received under the equivalent virtual array, which is composed of the power of the actual signal; represents the steering vector of the virtual array; represents the noise added to the virtual array, which can be expressed as 6. The low-complexity preprocessing method based on sparse array according to claim 2, characterized in that: Step 5 is as follows: divide the array obtained in step 4 into U+1 overlapping subarrays, each subarray has U+1 elements, and the position of the kth element in the i-th subarray in the original array is: {(i+kU-2)d|i,k=1,...,U+1}, the received data Y of the i-th subarray i (t) is Y i (t) = [y i-U-1 (t),…,y i-1 (t)] T , The received data of U+1 sub-arrays are used for Toeplitz reconstruction to construct the covariance matrix R Y Satisfy R Y =[Y1(t),Y2(t),…,Y U+1 (t)].

7. A method for estimating direction of arrival based on a low-complexity preprocessing method of a sparse array, characterized in that: The low-complexity preprocessing method based on sparse arrays as claimed in any one of claims 1 to 6 further comprises step 6, obtaining a signal subspace or a noise subspace according to the number of signal sources, and obtaining a signal arrival direction, specifically comprising the following steps: Step 6-1: recovering the covariance matrix of the rank Perform eigenvalue decomposition Among them, U represents the feature space, Σ represents the eigenvalue, and U s represents the signal space, U n represents the noise subspace, Σ s represents the eigenvalue corresponding to the signal subspace, Σ n Represents the eigenvalue corresponding to the noise subspace; Step 6-2: Using the orthogonality of the two subspaces, construct a search function to perform angle search. f MUSIC The θ corresponding to the Q maximum values ​​of (θ) is the arrival direction.

Images