Efficient DOA estimation algorithm based on expanded co-prime array

Abstract

The invention discloses an efficient DOA estimation algorithm based on an expanded co-prime array, and the algorithm comprises the steps: enabling K narrowband far-field information sources to transmit narrowband far-field signals at the same time, enabling the narrowband far-field signals to enter the expanded co-prime array, carrying out the T times of snapshot number sampling of the expanded co-prime array, and enabling the received signal vector of the expanded co-prime array obtained during the tth time of snapshot number sampling to be x (t); respectively calculating UnUnH and a (theta k); k roots zk are obtained by using a Root MUSIC algorithm; and calculating K initial DOA estimated values # imgabs0 # by combining an initial DOA angle estimation formula, and calculating an estimated direction of arrival of each narrowband far-field information source incident to the expanded co-prime array to obtain K estimated directions of arrival theta'k. As a receiving signal vector x (t) obtained by expanding a co-prime array cannot be directly applied to the Root MUSIC algorithm, the efficient DOA estimation algorithm disclosed by the invention can be directly applied to the Root MUSIC algorithm by constructing a selection matrix G to calculate and obtain a (theta k).

Description

Technical Field

[0001] The present invention belongs to the technical field of array signal processing, and particularly relates to an efficient DOA estimation algorithm based on an unfolded coprime array. Background Technique

[0002] Direction of Arrival (DOA) estimation is a key issue in the field of array signal processing and plays an important role in fields such as radar, sonar, communication, and seismology. Currently, most of the research on DOA estimation is proposed based on uniform linear arrays. Although the structure of the uniform linear array is simple, there are some disadvantages, such as a small array aperture, limited degrees of freedom (DOF), and low DOA estimation accuracy in a low signal-to-noise ratio (SNR) environment. With the in-depth research on array structures, non-uniform array structures such as minimum redundancy arrays, coprime arrays, and nested arrays have been proposed. Compared with the uniform linear array, these non-uniform array structures break through the half-wavelength limit of the element spacing. Therefore, they have a larger array aperture under the same number of array elements. Among the non-uniform array structures, the coprime array has a flexible array arrangement, low element mutual coupling, and high positioning and direction-finding accuracy, showing broad application potential. The unfolded coprime array (Huang X, Zhao X, Lu W. Joint frequency and DOA estimation of sub-Nyquist sampling multi-band sources with unfolded coprime arrays [J]. Multidimensional Systems and Signal Processing, 2022, 33(4): 1257-1272.) is an improved version of the coprime array. By arranging the two sub-arrays of the traditional coprime array in the reverse direction, the array aperture is further expanded, and there is a significant improvement in terms of increasing the degrees of freedom and positioning accuracy compared with the traditional coprime array.

[0003] Currently, many two-dimensional DOA estimation methods have been proposed by research scholars based on the unfolded coprime array. For example:

[0004] (1) A bistatic MIMO radar positioning technology based on an unfolded coprime array (Zhou Wei, Wang Qiang, Tang Jun, et al. Bistatic unfolded coprime array MIMO radar DOD, DOA dimension reduction estimation algorithm [J]. Journal of Nanjing University of Posts and Telecommunications: Natural Science Edition, 2020, 40(2): 20-26.). The above method uses the unfolded coprime array as the transceiver matrix to expand the array aperture and uses the MUSIC algorithm for DOA estimation, thus significantly improving the estimation performance, but the computational complexity is relatively large.

[0005] (2) A method for DOA estimation using a multiple signal classification algorithm based on an unfolded co-prime array structure (Zheng W, Zhang X, Gong P, et al. DOA estimation for coprime linear arrays: An ambiguity-free method involving full DOFs [J]. IEEE Communications Letters, 2017, 22(3): 562 - 565.). The above method utilizes its own information and mutual information to achieve full degrees of freedom and has better estimation performance, and is superior to the coprime linear array and the subarray-based method. However, this method has problems of wasting the uniformity of two subarrays and increasing the computational complexity due to involving a peak search step.

[0006] (3) A method for two-dimensional DOA angle estimation using an unfolded co-prime L-shaped array structure (Gong P, Zhang X, Zheng W. Unfolded coprime L-shaped arrays for two-dimensional direction of arrival estimation [J]. International journal of electronics, 2018, 105(9): 1501 - 1519.). The above method benefits from the extended array aperture and achieves more accurate elevation angle and azimuth angle estimation, but the problem of high computational complexity still cannot be avoided.

[0007] To reduce the computational complexity, the traditional Root MUSIC algorithm can be introduced, but the sparsity of the unfolded co-prime array causes the traditional Root MUSIC algorithm to be unable to be directly applied to the unfolded co-prime array (Peng Yiling, Zhou Jianjiang, Deng Yao, Huang Jinhai, Xu Zhe. DOA tracking algorithm based on OPASTd under unfolded co-prime array [J]. Modern Radar, 1 - 12.). Summary of the Invention

[0008] Aiming at the deficiencies of the prior art, the purpose of the present invention is to provide an efficient DOA estimation algorithm based on an unfolded co-prime array.

[0009] The present invention is realized through the following technical solutions.

[0010] An efficient DOA estimation algorithm based on an unfolded co-prime array, comprising the following steps:

[0011] S1. At the same moment, let K narrowband far-field sources each emit narrowband far-field signals incident on the expanded co-prime array, where k = 1, 2,..., K. The actual angle of arrival of the narrowband far-field signal emitted by the k-th narrowband far-field source incident on the expanded co-prime array is unknown. Perform T snapshots sampling on the expanded co-prime array. At each snapshots sampling, obtain the received signal vector of the expanded co-prime array, and the received signal vector is represented by a matrix containing the unknown angle of arrival; the received signal vector of the expanded co-prime array obtained at the t-th snapshots sampling is x(t).

[0012] S2. Calculate U based on x(t) n U n H , where U n is the noise subspace, is the conjugate transpose of the noise subspace U n ;

[0013] Calculate K a(θ k ). The method for calculating each a(θ k ) includes steps 1 to 2:

[0014] Step 1. Construct the selection matrix G;

[0015] Step 2. Let Substitute a(θ k )′ into the following formula to calculate a(θ k );

[0016] a(θ k ) = Ga(θ k )′

[0017] where j is the imaginary part of the complex number, e is the natural constant, λ is the wavelength of the narrowband far-field signal, D is the number of array elements of the complete uniform linear array. Assume that K narrowband far-field sources each emit narrowband far-field signals incident on the complete uniform linear array at the same moment, and perform T snapshots sampling on the complete uniform linear array. The narrowband far-field signal emitted by the k-th narrowband far-field source is incident on the complete uniform linear array at the angle of arrival of θ k , and the angle of arrival of the narrowband far-field signal emitted by the k-th narrowband far-field source incident on the complete uniform linear array and on the expanded co-prime array is the same; d is the element spacing in the complete uniform linear array;

[0018] S3. Calculate K roots z k , where, according to a(θ k ) and U n U n H Use the Root MUSIC algorithm to obtain each root z k ;

[0019] S4, based on K roots z k And the preliminary DOA angle estimation formula, calculate K preliminary DOA estimates Among them, the preliminary DOA angle estimation formula is:

[0020]

[0021] S5, based on the preliminary DOA estimation value, calculate the estimated arrival angle of each narrowband far-field source incident on the unfolded coprime array, and obtain K estimated arrival angles θ′1, θ′2, ..., θ K ′, where the estimated angle of arrival θ of the k-th narrowband far-field source incident on the unfolded coprime array is calculated k The method includes S5-1 and S5-2:

[0022] S5-1, obtain the steering vector matrix

[0023] Based on K preliminary DOA estimates Calculate K estimated direction vectors Make K estimated direction vectors form a steering vector matrix

[0024] S5-2, let the estimated arrival angle of the k-th narrowband far-field signal emitted by the narrowband far-field source incident on the unfolded coprime array be θ′ k , the narrowband far-field signal emitted by the kth narrowband far-field source is θ′ k The direction vector of the incident expanded coprime array is a(θ′ k ), by K a(θ′ k ) constitute the steering vector matrix A(θ′), A(θ′)=[a(θ′1), a(θ′2),..., a(θ′ K )];

[0025] Combining the following formulas, we can get the estimated angle of arrival θ′ k :

[0026]

[0027] In the formula, is the noise subspace U n The conjugate transpose of To perform a first-order Taylor expansion on the steering vector matrix A(θ').

[0028] In the above technical solution, the received signal vector x(t) is:

[0029] x(t)=A(θ)S(t)+N(t)

[0030] Where \(t = 1, 2,\cdots, T\), \(S(t)\) is a vector composed of \(K\) incident signals received by the expanded co-prime array at the \(t\)-th snapshot sampling, \(A(\theta)\) is the steering vector matrix, and the dimension of \(A(\theta)\) is the number of array elements \(M\) of the expanded co-prime array \(\times\) the number of narrowband far-field signal sources \(K\); \(N(t)\) is the additive white Gaussian noise vector received by the expanded co-prime array at the \(t\)-th snapshot sampling.

[0031] The expanded co-prime array includes sub-array 1 and sub-array 2. The number of array elements of sub-array 1 is \(N_1\), the number of array elements of sub-array 2 is \(N_2\), \(d=\lambda / 2\), and \(\lambda\) is the wavelength of the narrowband far-field signal.

[0032] In the above technical solution, calculate \(U\) n U n H The method includes ① - ③:

[0033] ① Randomly select the received signal vector \(x(t)\) of the expanded co-prime array obtained at the \(t\)-th snapshot sampling and calculate the covariance matrix \(R\) according to it:

[0034] \(R = E\{x(t)x\) H (t)\}

[0035] Where \(E\{\cdot\}\) represents calculating the expectation, and \(x\) H (t)\) is the conjugate transpose of the received signal vector \(x(t)\) of the expanded co-prime array;

[0036] ② Perform eigenvalue decomposition on the covariance matrix \(R\) to obtain \(w\) eigenvalues, and sort the \(w\) eigenvalues in descending order to obtain the sorted eigenvalue sequence \(\lambda_1,\lambda_2,\cdots,\lambda\) w ; Let the eigenvalues \(\lambda_1\) to \(\lambda\) in the sorted eigenvalue sequence K form the diagonal matrix \(\Sigma\) s , and let the eigenvalues \(\lambda\) K+1 to \(\lambda\) w in the sorted eigenvalue sequence form the diagonal matrix \(\Sigma\) n ; Based on the diagonal matrix \(\Sigma\) s and the diagonal matrix \(\Sigma\) n obtain the following formula:

[0037]

[0038] Where \(U\) s is the signal subspace, and \(U\) n is the noise subspace;

[0039] ③ Perform a transformation on to obtain the following formula:

[0040]

[0041] In the formula, is the variance of the additive white Gaussian noise received by the expanded co-prime array during the sampling of the number of snapshots for this time; m is the high-order power;

[0042] Let m approach infinity. Based on obtain U when m approaches infinity n U n H where λ k is any one of the eigenvalues from eigenvalue λ1 to eigenvalue λ K in.

[0043] In the above technical solution, the method for constructing the selection matrix G is as follows: only one element in each row of the selection matrix G is a non-zero element, and the remaining elements are zero, and all non-zero elements in the selection matrix G are 1; set the array Array = [1, N2 + 1,..., N2(N1 - 1) + 1,..., D], D = N2(N1 - 1) + N1(N2 - 1) + 1. In the array Array, the values of the elements between 1 and N2(N1 - 1) + 1 increase, and the difference between adjacent elements is N2. The values of the elements between N2(N1 - 1) + 1 and D increase, and the difference between adjacent elements is N1; the non-zero element in the i-th row of the selection matrix G is in the z-th column of the selection matrix G, where z is the value of the i-th element in the array Array, and i = 1,..., M; M = N1 + N2 - 1.

[0044] In the above technical solution, the dimension of the selection matrix G is M×D.

[0045] In the above technical solution, a(θ k ) is a column vector with a dimension of M×1.

[0046] a(θ k )′ is a column vector, the dimension of a(θ k )′ is D×1, D = N2(N1 - 1) + N1(N2 - 1) + 1, and each element in a(θ k )′ is the direction vector of an element in the complete uniform linear array when the narrowband far-field signal emitted by the k-th narrowband far-field source is incident.

[0047] In the above technical solution, the i'-th element in is i' = 1, 2,..., M, set the array Arr, Arr = [p1, p1,..., p M= [0, N2, ..., N2(N1 - 1), ..., N2(N1 - 1) + N1(N2 - 1)], the values of the elements between 0 and N2(N1 - 1) in the array Arr increase, and the difference between adjacent elements is N2. The values of the elements between N2(N1 - 1) and N2(N1 - 1) + N1(N2 - 1) in the array Arr increase, and the difference between adjacent elements is N1.

[0048] The efficient DOA estimation algorithm of the present invention solves the problem that the traditional Root MUSIC algorithm cannot be directly applied due to the sparsity of the expanded co-prime array by introducing a selection matrix. The efficient DOA estimation algorithm of the present invention has high parameter estimation accuracy and lower complexity compared with the MUSIC algorithm. Description of the Drawings

[0049] Figure 1 It is the comparison graph of the estimated arrival angle of the wave θ' obtained in Example 2 k and the actual arrival angle of the wave;

[0050] Figure 2 It is the comparison graph of the complexity of the efficient DOA estimation algorithm of the present invention and the complexity of the MUSIC method in Example 3;

[0051] Figure 3 It is the comparison graph of the complexity of the efficient DOA estimation algorithm of the present invention and the complexity of the MUSIC method in Example 4. Detailed Embodiment

[0052] The technical solution of the present invention will be further described in detail below in conjunction with the drawings and embodiments.

[0053] Example 1

[0054] An efficient DOA estimation algorithm based on an expanded co-prime array includes the following steps:

[0055] S1, Let K narrowband far-field signal sources each emit narrowband far-field signals at the same moment and incident on the expanded co-prime array, let k = 1, 2, …, K, the actual arrival angle of the narrowband far-field signal emitted by the k-th narrowband far-field signal source incident on the expanded co-prime array is unknown, perform T snapshots on the expanded co-prime array, and obtain the received signal vector of the expanded co-prime array during each snapshot. The received signal vector is represented by a matrix containing the unknown arrival angle of the wave. The received signal vector of the expanded co-prime array obtained during the t-th snapshot is x(t);

[0056] Among them, the received signal vector x(t) is:

[0057] x(t) = A(θ)S(t) + N(t) (1)

[0058] where \(t = 1, 2, \cdots, T\), \(S(t)\) is a vector composed of \(K\) incident signals received by the expanded co-prime array at the \(t\)-th snapshot sampling, \(A(\theta)\) is the steering vector matrix, and the dimension of \(A(\theta)\) is the number of array elements \(M\) of the expanded co-prime array \(\times\) the number of narrowband far-field signal sources \(K\); \(N(t)\) is the additive white Gaussian noise vector received by the expanded co-prime array at the \(t\)-th snapshot sampling;

[0059] The narrowband far-field signals emitted by each of the \(K\) narrowband far-field signal sources are mutually uncorrelated; the expanded co-prime array includes sub-array 1 and sub-array 2. Both sub-array 1 and sub-array 2 are sparse linear arrays. The element spacing of sub-array 1 is \(d_1 = N_2d\), the number of elements of sub-array 1 is \(N_1\), the element spacing of sub-array 2 is \(d_2 = N_1d\), the number of elements of sub-array 2 is \(N_2\), \(d\) is the unit spacing, \(d=\lambda / 2\), \(\lambda\) is the wavelength of the narrowband far-field signal, \(N_1\) and \(N_2\) are co-prime, and sub-array 1 and sub-array 2 share a common element (the common element is the reference element, the reference element is the last element of sub-array 1, and at the same time the reference element is also the first element of sub-array 2);

[0060] S2, calculate \(U\) respectively n U n H and \(a(\theta\) k )

[0061] Calculate \(U\) n U n H The method of calculating \(U\) includes ① - ③:

[0062] ① Randomly select the received signal vector \(x(t)\) of the expanded co-prime array obtained at the \(t\)-th snapshot sampling and calculate the covariance matrix \(R\) according to it:

[0063] \(R = E\{x(t)x\) H (t)\} (2)

[0064] where \(E\{\cdot\}\) represents calculating the expectation, and \(x\) H (t)\) is the conjugate transpose of the received signal vector \(x(t)\) of the expanded co-prime array;

[0065] ② Perform eigenvalue decomposition on the covariance matrix \(R\) in formula (2) to obtain \(w\) eigenvalues, and sort the \(w\) eigenvalues in descending order to obtain the sorted eigenvalue sequence \(\lambda_1, \lambda_2, \cdots, \lambda\) w ; make the eigenvalues \(\lambda_1\) to \(\lambda\) in the sorted eigenvalue sequence K form the diagonal matrix \(\Sigma\) s , make the eigenvalues \(\lambda\) K+1 to \(\lambda\) w in the sorted eigenvalue sequence form the diagonal matrix \(\Sigma\) n ; based on the diagonal matrix \(\Sigma\) sand the diagonal matrix Σ n Convert formula (2) into formula (3):

[0066]

[0067] where U s is the signal subspace, and U n is the noise subspace;

[0068] ③ Transform formula (3) to obtain formula (4):

[0069]

[0070] where is the variance of the additive white Gaussian noise received by the expanded co-prime array during this snapshot sampling; m is the high-order power;

[0071] According to formula (4) and (λ k is any one of the eigenvalues λ1 to eigenvalue λ K ), obtain (when m approaches infinity, make ), and obtain U when m approaches infinity n U n H and the noise subspace U n ;

[0072] The method for calculating a(θ k ) includes steps 1 to 2:

[0073] Step 1: Construct the selection matrix G. The dimension of the selection matrix G is M×D. The method for constructing the selection matrix G (Zhang Q, Li J, Tang Y, et al. An efficient 2-D DOA estimation method for unfolded coprime L-shaped array[J]. IEEE Systems Journal, 2022, 17(2): 3328-3331.) is as follows: In each row of the selection matrix G, only one element is non-zero and the remaining elements are zero, and all non-zero elements in the selection matrix G are 1. Set the array Array = [1, N2 + 1,..., N2(N1 - 1) + 1,..., D], where D = N2(N1 - 1) + N1(N2 - 1) + 1. The unfolded coprime array includes subarray 1 and subarray 2. N1 is the number of array elements in subarray 1, and N2 is the number of array elements in subarray 2. In the array Array, the values of the elements between 1 and N2(N1 - 1) + 1 increase and the difference between adjacent elements is N2, and the values of the elements between N2(N1 - 1) + 1 and D increase and the difference between adjacent elements is N1. The non-zero element in the i-th row of the selection matrix G is in the z-th column of the selection matrix G, where z is the value of the i-th element in the array Array, and i = 1,..., M. M is the number of array elements of the unfolded coprime array, and M = N1 + N2 - 1.

[0074] Step 2: Let Substitute a(θ k )′ into formula (5) and calculate to obtain a(θ k ).

[0075] a(θ k ) = Ga(θ k )′ (5)

[0076] where j is the imaginary part of the complex number, and e is the natural constant. Assume that K narrowband far-field signal sources each emit narrowband far-field signals at the same time and are incident on the complete uniform linear array, and T snapshots are sampled for the complete uniform linear array. The k-th narrowband far-field signal source emits a narrowband far-field signal and is incident on the complete uniform linear array at the angle of arrival θ k . The angle of arrival of the narrowband far-field signal emitted by the k-th narrowband far-field signal source incident on the complete uniform linear array is the same as that incident on the unfolded coprime array. d is the element spacing in the complete uniform linear array.

[0077] a(θ k ) is a column vector with dimension M×1, where M is the number of array elements of the unfolded coprime array, and M = N1 + N2 - 1.

[0078] a(θ k )′ is a column vector, and a(θ k)′ has a dimension of D×1, where D is the number of array elements in a full uniform linear array, and D = N2(N1 - 1) + N1(N2 - 1) + 1, a(θ k )′ Each element in is the direction vector of the narrowband far-field signal emitted by the k-th narrowband far-field source incident on an array element in the full uniform linear array.

[0079] Since the received signal vector x(t) of the co-prime array cannot be directly applied to the traditional Root MUSIC algorithm by expansion, a(θ k ) is calculated through constructing a selection matrix G and can be directly applied to the traditional Root MUSIC algorithm, solving the problem that the traditional Root MUSIC algorithm cannot be directly applied to non-uniform arrays due to the sparsity of the co-prime array after expansion.

[0080] S3. Calculate K roots z k , and the calculation method of each root z k includes: According to a(θ k ) and U n U n H Use the Root MUSIC algorithm (Rao B D, Hari K V S. Performance analysis of root-MUSIC[J]. IEEE Transactions on Acoustics, Speech, and Signal Processing, 1989, 37(12): 1939 - 1949.) to obtain the root z k ;

[0081] S4. According to the K roots z k obtained in S3, combined with the preliminary DOA angle estimation formula, calculate the preliminary DOA estimation value of the narrowband far-field signal emitted by each narrowband far-field source incident on the co-prime array after expansion The K preliminary DOA estimation values are Among them, the preliminary DOA angle estimation formula:

[0082]

[0083] In the formula, λ is the wavelength of the narrowband far-field signal;

[0084] S5. According to the preliminary DOA estimation values, calculate the estimated arrival angles of each narrowband far-field source incident on the co-prime array after expansion to obtain K estimated arrival angles θ′1, θ′2,..., θ K ′, where the method for calculating the estimated arrival angle θ k ′ of the k-th narrowband far-field source incident on the co-prime array after expansion includes S5-1 and S5-2:

[0085] S5-1, Obtain the steering vector matrix

[0086] According to the K initial DOA estimation values Calculate K estimated direction vectors The i'-th element in is

[0087] i' = 1, 2,......, M, set the array Arr, Arr = [p1, p1,..., p M = [0, N2,..., N2(N1 - 1),..., N2(N1 - 1)+N1(N2 - 1)], the values of the elements between 0 and N2(N1 - 1) in the array Arr increase and the difference between adjacent elements is N2, the values of the elements between N2(N1 - 1) and N2(N1 - 1)+N1(N2 - 1) in the array Arr increase and the difference between adjacent elements is N1, j is the imaginary part of the complex number, and e is the natural constant;

[0088] Make the K estimated direction vectors form the steering vector matrix

[0089] S5-2, Let the estimated arrival angle of the narrowband far-field signal emitted by the k-th narrowband far-field source incident on the expanded co-prime array be θ' k , the narrowband far-field signal emitted by the k-th narrowband far-field source with θ' k incident on the direction vector of the expanded co-prime array is a(θ' k ), and the steering vector matrix A(θ') (A(θ') is an unknown quantity) is composed of K a(θ' k ), A(θ') = [a(θ'1), a(θ'2),..., a(θ' K )];

[0090] Since under ideal conditions, the steering vector matrix A(θ') is orthogonal to the noise subspace U n , the formula (7) can be obtained;

[0091]

[0092] In the formula, is the conjugate transpose of the noise subspace U n ;

[0093] Perform the first-order Taylor expansion on the steering vector matrix A(θ') according to the formula (8);

[0094]

[0095] Combining formula (7) and formula (8), we can get the estimated angle of arrival θ′ k .

[0096] Example 2

[0097] On the basis of Example 1, an efficient DOA estimation algorithm based on an unfolded coprime array is provided, wherein K=3, N1=4, N2=5, and the actual arrival angle of the narrowband far-field signal transmitted by the K narrowband far-field sources and incident on the unfolded coprime array is [θ1, θ2, θ3]=[10°, 20°, 30°]; d1=5d, d2=4d, d=0.5λ(λ=1), and the power of all narrowband far-field sources is equal, and the signal-to-noise ratio SNR is The number of snapshots T = 500, the high-order power m is 5, where Represents the signal power of a narrowband far-field source;

[0098] Actual arrival angle and estimated arrival angle θ′ k like Figure 1 As shown by Figure 1 It can be seen that the θ′ obtained by the efficient DOA estimation algorithm of the present invention is k , under various test conditions, the error is controlled within a very small range, which is significantly better than traditional algorithms and shows extremely high reliability and accuracy.

[0099] Example 3

[0100] On the basis of Example 1, the computational complexity of the efficient DOA estimation algorithm of the present invention is compared with the MUSIC method (Zheng W, Zhang X, Gong P, et al. DOA estimation for coprime linear arrays: An ambiguity-free method involving full DOFs [J]. IEEE Communications Letters, 2017, 22 (3): 562-565.), and the computational complexity of the efficient DOA estimation algorithm of the present invention and the computational complexity of the MUSIC method are obtained. Figure 2 As shown, the ordinate is complexity and the abscissa is the number of array elements N1;

[0101] The parameters of the efficient DOA estimation algorithm of the present invention are set as follows: let K = 3, N1 = an integer in the range of 12 to 21, N2 = 11, the number of snapshots T = 500, the high-order power m is 5, the power of all narrowband far-field sources is equal, and the signal-to-noise ratio SNR is in, Represents the signal power of a narrowband far-field source;

[0102] Example 4

[0103] On the basis of Embodiment 1, the computational complexity of the efficient DOA estimation algorithm of the present invention is compared with that of the MUSIC method, and the computational complexity of the efficient DOA estimation algorithm of the present invention and the computational complexity of the MUSIC method are obtained. As Figure 3 shown, the ordinate is the complexity, and the abscissa is the number of array elements N2; wherein, the parameter settings of the efficient DOA estimation algorithm of the present invention are: let K = 3, N1 = 13, N2 be an integer in the range of 20 to 30, the number of snapshots T = 500, the high-order power m is 5, the powers of all narrowband far-field signal sources are equal, and the signal-to-noise ratio SNR is wherein, represents the signal power of the narrowband far-field signal source;

[0104] From Figure 2 and Figure 3 it can be seen that the efficient DOA estimation algorithm of the present invention has a lower computational complexity compared to the spectral peak search algorithm (MUSIC).

[0105] Among them, the calculation formula for the complexity of the efficient DOA estimation algorithm of the present invention:

[0106] O[(N1 + N2 - 1) 2 T + 2(2N1N2 - (N1 + N2) 3 .

[0107] The calculation formula for the complexity of the MUSIC method:

[0108] O[(N1 + N2 - 1) 2 T + (N1 + N2 - 1) 3 + (180 / Δα)(N1 + N2 - 1) 2 .

[0109] Among them, Δα represents the search step size, and the search step size is 0.1°.

[0110] In summary, the accuracy of the estimated angle of arrival calculated by the efficient DOA estimation algorithm of the present invention is close to 100%, and its computational complexity is lower than that of the spectral peak search algorithm (MUSIC).

[0111] The above is an exemplary description of the present invention. It should be noted that without departing from the core of the present invention, any simple deformation, modification, or equivalent replacement that can be made by those skilled in the art without creative labor falls within the protection scope of the present invention.

Claims

1. An efficient DOA estimation algorithm based on unfolded coprime arrays, characterized in that: The following steps are involved: S1, make K narrowband far-field signal sources transmit narrowband far-field signals to the expanded coprime array at the same time, perform T snapshot number sampling on the expanded coprime array, and obtain the received signal vector of the expanded coprime array at each snapshot number sampling; the received signal vector of the expanded coprime array obtained at the t-th snapshot number sampling is x(t), t=1, 2, ..., T; S2, calculate U based on x(t) n U n H , where U n is the noise subspace, is the noise subspace U n The conjugate transpose of ; Calculate K a(θ k ), calculate each a(θ k ) method includes: k )′ into the following formula to calculate a(θ k ); a(θ k )=Ga(θ k )′ Where G is the selection matrix, and the kth narrowband far-field source transmits a narrowband far-field signal with θ k The wave arrival angle incident on the complete uniform linear array, k = 1, 2, ..., K, θ k The same as the arrival angle of the kth narrowband far-field source transmitting the narrowband far-field signal incident on the unfolded coprime array, a(θ k )′ is a column vector with dimension D×1, a(θ k )′, each element is the direction vector of the narrow-band far-field signal emitted by the k-th narrow-band far-field source incident on an array element in the complete uniform linear array, and D is the number of array elements in the complete uniform linear array; S3, calculate K roots z k , where, according to a(θ k ) and U n U n H Use the Root MUSIC algorithm to get each root z k ; S4, based on K roots z k And the preliminary DOA angle estimation formula, calculate K preliminary DOA estimation values: Among them, the preliminary DOA angle estimation formula is: S5, based on the preliminary DOA estimation value, calculate the estimated arrival angle of each narrowband far-field source incident on the unfolded coprime array, and obtain K estimated arrival angles θ1′, θ2′, …, θ K ′.

2. The efficient DOA estimation algorithm according to claim 1, characterized in that: The actual arrival angle of the narrowband far-field signal emitted by the k-th narrowband far-field source and incident on the unfolded coprime array is unknown, and the received signal vector is represented by a matrix containing the unknown arrival angle.

3. The efficient DOA estimation algorithm according to claim 1, characterized in that: The received signal vector x(t) is: x(t)=A(θ)S(t)+N(t) Where S(t) is a vector composed of K incident signals received by the unfolded coprime array at the t-th snapshot sampling, A(θ) is a steering vector matrix, and the dimension of A(θ) is the number of array elements of the unfolded coprime array M × the number of narrow-band far-field signal sources K; N(t) is the additive white Gaussian noise vector received by the unfolded coprime array at the t-th snapshot sampling.

4. The efficient DOA estimation algorithm according to claim 1, characterized in that: The expanded coprime array includes sub-array 1 and sub-array 2, the number of array elements of sub-array 1 is N1, and the number of array elements of sub-array 2 is N2; d=λ / 2.

5. The efficient DOA estimation algorithm according to claim 1, 2 or 3, characterized in that: Calculate U n U n H The methods include ①~③: ① Randomly select the received signal vector x(t) of the unfolded coprime array obtained when sampling the t-th snapshot number and calculate the covariance matrix R based on it: R=E{x(t)x H (t)} In the formula, E{·} represents the expected value, x H (t) is the conjugate transpose of the received signal vector x(t) of the unfolded coprime array; ② Perform eigendecomposition on the covariance matrix R to obtain w eigenvalues, and sort the w eigenvalues ​​in descending order to obtain the sorted eigenvalue sequence λ1, λ2, ..., λ w ; Make the eigenvalue λ1~eigenvalue λ in the sorted eigenvalue sequence K Form a diagonal matrix Σ s , so that the eigenvalue λ in the sorted eigenvalue sequence K+1 ~Eigenvalue λ w Form a diagonal matrix Σ n ; Based on the diagonal matrix Σ s and the diagonal matrix Σ n The following formula is obtained: Where U s is the signal subspace, U n is the noise subspace; ③Yes After transformation, we get the following formula: In the formula, is the variance of the additive white Gaussian noise received by the coprime array when sampling the snapshot number; m is a high-order power; Make m infinite, based on Find U when m is infinite n U n H , where λ k is the eigenvalue λ1 to the eigenvalue λ K Any eigenvalue in .

6. The efficient DOA estimation algorithm according to claim 4, characterized in that: The method for constructing the selection matrix G is as follows: only one element in each row of the selection matrix G is a non-zero element and the remaining elements are zero, and all non-zero elements in the selection matrix G are 1; set array Array = [1, N2+1, ..., N2(N1-1)+1, ..., D], D = N2(N1-1)+N1(N2-1)+1, in the array Array, the values ​​of the elements between 1 and N2(N1-1)+1 increase and the difference between adjacent elements is N2, the values ​​of the elements between N2(N1-1)+1 and D increase and the difference between adjacent elements is N1; the non-zero elements in the i-th row of the selection matrix G are in the z-th column of the selection matrix G, z is the value of the i-th element in the array Array, i = 1, ..., M; M = N1+N2-1, and the dimension of the selection matrix G is M×D.

7. The efficient DOA estimation algorithm according to claim 1, characterized in that: Calculate the estimated angle of arrival θ of the kth narrowband far-field source incident on the unfolded coprime array k The method includes S5-1 and S5-2: S5-1, obtain the steering vector matrix Based on K preliminary DOA estimates Calculate K estimated direction vectors Make K estimated direction vectors form a steering vector matrix S5-2, let the estimated arrival angle of the k-th narrowband far-field signal emitted by the narrowband far-field source incident on the unfolded coprime array be θ′ k , the narrowband far-field signal emitted by the kth narrowband far-field source is θ′ k The direction vector of the incident expanded coprime array is a(θ′ k ), by K a(θ′ k ) constitute the steering vector matrix A(θ′), A(θ′)=[a(θ′1), a(θ′2),..., a(θ′ K )]; Combining the following formulas, we can get the estimated angle of arrival θ′ k : In the formula, is the noise subspace U n The conjugate transpose of It is obtained by performing a first-order Taylor expansion on the steering vector matrix A(θ′).

8. The efficient DOA estimation algorithm according to claim 1, characterized in that: a(θ k ) is a column vector of dimension M×1; make j is the imaginary part of the complex number, e is a natural constant, λ is the wavelength of the narrowband far-field signal, and d is the element spacing in the complete uniform linear array.

9. The efficient DOA estimation algorithm according to claim 4, characterized in that: D=N2(N1-1)+N1(N2-1)+1.

10. The efficient DOA estimation algorithm according to claim 7, characterized in that: The i′th element in Set the array Arr, Arr = [p1, p1, ..., p M ]=[0, N2, ..., N2(N1-1), ..., N2(N1-1)+N1(N2-1)], the values ​​of the elements between 0 and N2(N1-1) in the array Arr are increasing and the difference between adjacent elements is N2, the values ​​of the elements between N2(N1-1) and N2(N1-1)+N1(N2-1) in the array Arr are increasing and the difference between adjacent elements is N1, j is the imaginary part of the complex number, e is a natural constant, λ is the wavelength of the narrowband far-field signal, and d is the element spacing in the complete uniform linear array.

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