Coherent signal direction-of-arrival estimation method for co-prime array matrix reconstruction

Abstract

The invention discloses a coherent signal direction-of-arrival estimation method for co-prime array matrix reconstruction. According to the method, a co-prime array is expanded into a conjugate augmented virtual array by using the conjugate symmetry characteristic of a cross-correlation function between array element receiving signals with different time lengths. Zero padding is carried out at the hole position of the conjugate augmented virtual array by utilizing an array interpolation thought, so that a virtual uniform linear array is formed; then, on the basis of the virtual uniform linear array, a space-time correlation matrix is constructed by adopting a matrix reconstruction method and utilizing strong correlation and noise weak correlation of narrow-band signals under different snapshots, and the influence of noise is eliminated; and then the information of the space-time correlation matrix is utilized to reconstruct the Toeplitz matrix through an MTOEP method, so that the energy of the signal is effectively improved, and the signal-to-noise ratio is further improved. And finally, estimating the DOA of the signal by using a TLS-ESPRIT method. The method provided by the invention has good estimation performance and relatively high resolution under the conditions of low signal-to-noise ratio and few snapshots.

Description

Technical Field

[0001] The present invention belongs to the field of sensor array signal processing technology in the field of signal and information processing, and relates to a method for estimating the direction of arrival of coherent signals reconstructed by a coprime array matrix. Specifically, in a coherent signal scenario, a traditional coprime array is expanded into a virtual uniform linear array by conjugate augmentation and array interpolation, and then a Toeplitz matrix is ​​reconstructed using all rows of a spatiotemporal correlation matrix to realize the method for estimating the direction of arrival of coherent signals of a coprime array. Background Art

[0002] Direction of Arrival (DOA) estimation is one of the hot research issues in the field of array signal processing in recent years, and has broad application prospects in the fields of radar, sonar, wireless communication, etc. DOA estimation is an important component of smart antennas and is also of great significance in wireless positioning applications. In the past decade or so, researchers at home and abroad have proposed many high-resolution coprime array DOA estimation algorithms. On the one hand, in order to take advantage of the degree of freedom provided by coprime arrays, DOA estimation can be performed by constructing an enhanced virtual array and realizing its equivalent virtual array signal statistics. Since the derived virtual array contains more virtual sensors, the degree of freedom limitation of the number of physical sensors is overcome. On the other hand, in the azimuth estimation problem of coherent signals, spatial smoothing technology and Toeplitz matrix reconstruction are more popular. Spatial smoothing technology effectively realizes the resolution of coherent signals, but at the same time loses the effective aperture of the array and reduces the performance of direction of arrival estimation; the Toeplitz matrix reconstruction algorithm reconstructs the Toeplitz matrix using the row vectors of the array output covariance matrix. However, the Toeplitz matrix reconstruction method based on the arbitrary row vectors of the array output covariance matrix has signal correlation problems and poor robustness. However, the above two methods are mostly applicable to uniform linear arrays and are rarely used in coprime arrays.

[0003] This patent targets the existing coprime array DOA estimation method. First, the coprime array is expanded into a virtual array using conjugate augmentation, and then the holes of the non-uniform virtual array are filled with zeros to form a virtual uniform linear array. Then, the coherence of the narrowband signal and the correlation of uncorrelated noise at different snapshots are used to construct the spatiotemporal correlation matrix, which effectively eliminates the influence of noise. Then, the Toeplitz matrix is ​​reconstructed using each row of the spatiotemporal correlation matrix, which effectively increases the energy of the signal and thus improves the signal-to-noise ratio. Finally, it is combined with a subspace-based method to obtain the direction of arrival. When the number of snapshots is small, it has better DOA estimation performance and higher resolution. Summary of the invention

[0004] The purpose of the present invention is to address the shortcomings of the existing traditional coprime array DOA estimation method and propose a coherent signal direction of arrival estimation method based on coprime array matrix reconstruction. The method first uses conjugate augmentation and array interpolation to transform the coprime array into a virtual uniform linear array, then uses the strong correlation of narrowband signals and the weak correlation of noise to construct a spatiotemporal correlation matrix, and uses each row of the spatiotemporal correlation matrix to reconstruct a new Toeplitz matrix, and finally uses the TLS-ESPRIT method to obtain the DOA estimation result.

[0005] To achieve the above object, the present invention adopts the following technical solutions:

[0006] A coherent signal direction of arrival estimation method based on coprime array matrix reconstruction is proposed. Subarray 1 of the coprime array has M array elements, and the array element spacing is Nd. Subarray 2 has N array elements, and the array element spacing is Md. M and N are coprime integers, and M<N, d=λ / 2, λ is half wavelength, and the total number of array elements is L=M+N-1. The array element position set of the coprime array is expressed as There are K coherent signals from different angles θ=[θ1,θ2,…,θ K ] is incident on the coprime array, the number of snapshots is J, the first signal s1(t) is used as the reference signal, and the kth coherent signal is Among them, β k represents the signal amplitude, Δψ k =ωt represents s k The phase difference between s1(t) and s1(t), ω represents the signal frequency, and the received signal of the pth array element is expressed as Among them, n p (t) represents additive noise, l p Representing a collection The pth element of , l1 = 0, the coprime array received signal is expressed as in, represents the coherent signal vector, A represents the array manifold matrix, l p , p=1,…,M+N-1 represents the array element position, where the superscript “T” represents the transposition operation, n(t) represents the additive Gaussian white noise, and the power is And it is not related to the signal. The method for estimating the direction of arrival includes the following steps:

[0007] Step 1: The cross-correlation function of the received signal of the first array element and the received signal of the pth array element in the coprime array with respect to the time length τ (τ≠0) is expressed as in, Further expressed as Constructing vectors We get R(τ) = AR s (τ), according to R s (τ)=(R s (-τ) * , we get (R(-τ)) * =A * R s (τ), let the matrix A (2) and vector R (2) (τ) represents the 2nd to M+N-1th rows of the matrix A and the vector R(τ), respectively. (2) (τ) = A (2) R s (τ);

[0010] Step 2: According to the (R(-τ)) obtained in step 1 * and R (2) (τ), construct the matrix in, is the received signal value of a virtual array, is the direction matrix of the virtual array, is a single snapshot signal vector composed of K virtual incident signals, where the kth virtual incident signal is expressed as k=1,2,…,K, virtual array position set express The mirror symmetric set about the origin, the total number of array elements

[0011] Step 3: Based on the results obtained in step 2 Constructing pseudo snapshot data matrix in, is the virtual received signal value of the virtual array, T s is the pseudo sampling interval, the pseudo snapshot number J w = J / 2, is the pseudo signal matrix;

[0012] Step 4: Construct a uniform linear array set in Representing a collection The maximum value of , using interpolation, the sparse virtual array obtained in step 3 is transformed into a virtual uniform linear array, that is, the interpolated virtual array, and the corresponding data matrix is ​​expressed as in Representing a collection Contains but the set It is not included in Among them A z A steering matrix representing the interpolated virtual array;

[0013] Step 5: Order Use z(t) from step 4 to construct a (Z+1)×(Z+1) Toeplitz matrix in, Indicated by The Toeplitz matrix formed by Indicated by The Toeplitz matrix formed;

[0014] Step 6: According to the formula Construct a spatiotemporal correlation matrix, where the superscript "H" represents the Hermitian transpose operation. represents the array steering matrix, and the kth signal steering vector is Represents S(t) and the signal obtained by the i-th array element The correlation matrix between them, where the signal diagonal matrix S(t) = diag{R s (T s ),R s (2T s ),…,R s (J w T s )};

[0015] Step 7: According to formula s k (t-τ)≈s k (t)e -jωτ , the spatiotemporal correlation matrix obtained in step 6 Further expressed as

[0016] Step 8: According to the formula Reconstruct the Toeplitz matrix,

[0018] Step 9: According to the formula get Where J represents a switching matrix, which has 1 on the anti-diagonal and 0 elsewhere;

[0019] Step 10: According to the formula Pair Matrix Eigenvalue decomposition, where e i represents the eigenvector corresponding to the eigenvalue,

[0020] Step 11: Take The K largest eigenvalues ​​among the eigenvalues ​​constitute the signal subspace and are divided into U X and U Y Two parts;

[0021] Step 12: Construct the matrix And calculate the eigenvalue decomposition;

[0022] Step 13: Decompose U into K×K sub-matrices

[0023] Step 14: According to the formula Calculate eigenvalues

[0024] Step 15: According to the formula Calculate the coherent signal angle estimation value. Compared with the prior art, the technical solution of the present invention has the following technical effects:

[0025] (1) The present invention expands the sparse linear array into a virtual uniform linear array, thereby improving the resolution and estimation accuracy;

[0026] (2) The present invention uses the coherence of narrowband signals and the correlation of uncorrelated noise at different snapshots to construct a spatiotemporal correlation matrix, which effectively eliminates the influence of noise and can well distinguish the coherent signal source;

[0027] (3) The present invention has good robustness and high DOA estimation performance for coherent signals;

[0028] (4) The present invention still has good estimation performance when there are few snapshots. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 : Array model of this system;

[0030] Figure 2 : The conjugate augmented virtual array in the process of the patented signal processing method;

[0031] Figure 3 : Virtual uniform linear array after interpolation and zero filling in the signal processing method of this patent;

[0032] Figure 4 :The DOA estimation results of the patented signal processing method under three coherent signal sources;

[0033] Figure 5 : The relationship curve between the root mean square error and the signal-to-noise ratio of the patented signal processing method;

[0034] Figure 6 : The relationship curve between the success probability and signal-to-noise ratio of the patented signal processing method;

[0035] Figure 7 : The relationship curve between the root mean square error and the number of snapshots of the patented signal processing method;

[0036] Figure 8: The relationship curve between the success probability and the number of snapshots of the patented signal processing method; DETAILED DESCRIPTION

[0037] The present invention will now be further described with reference to the embodiments and the accompanying drawings:

[0038] The first embodiment: Figure 1 The array model of the coprime array used in the present invention is given. The coprime array consists of subarray 1 and subarray 2. Subarray 1 has 3 array elements, i.e., M=3, and the array element spacing is Nd=5d. Subarray 2 has 5 array elements, i.e., N=5, and the array element spacing is Md=3d, where d=0.5m. Except for the first array element, the other array elements of the coprime array will not overlap with each other, and the total number of array elements is L=M+N-1=7. The number of coherent signals is K=3, and the coherence coefficient (1,e jπ / 6 ,e jπ / 8 )The arrival angle is (0°, 21.5°, 50°), the signal-to-noise ratio is 10dB, and the number of snapshots is J=1000.

[0039] The above conditions are used to estimate the direction of arrival. The specific implementation process is as follows:

[0040] Taking the first signal s1(t) as the reference signal, the kth coherent signal is Among them, β k represents the signal amplitude, Δψ k =ωt represents s k The phase difference between s1(t) and s1(t), ω represents the signal frequency, and the received signal of the pth array element is expressed as Among them, n p (t) represents additive noise, l p Representing a collection The pth element of , l1 = 0, the coprime array received signal is expressed as Where s(t)=[s1(t),s2(t),…,s K (t)] T represents the coherent signal vector, K=3 represents the number of signals, A represents the array manifold matrix, l i , i=1,…,7 represents the array element position, where the superscript “T” represents the transposition operation, the signal label k=1,2,3, n(t) represents additive white Gaussian noise, and the power is and is not related to the signal;

[0041] Step 1: The cross-correlation function of the received signal of the first array element and the received signal of the pth array element in the coprime array with respect to the time length τ (τ≠0) is expressed as in, Further expressed as Constructing vectors We get R(τ) = AR s (τ), according to Get (R (-τ)) * =A * R s (τ), let the matrix A (2) and vector R (2) (τ) represents the 2nd to L=M+N-1=7th rows of the matrix A and the vector R(τ), respectively. (2) (τ) = A (2) R s (τ);

[0043] Step 2: According to the (R(-τ)) obtained in step 1 * and R (2) (τ), construct the matrix in, is the received signal value of a virtual array, is the direction matrix of the virtual array, is a single snapshot signal vector composed of K virtual incident signals, where the kth virtual incident signal is expressed as , k = 1, 2, 3, virtual array position set express The mirror symmetric set about the origin, the total number of array elements

[0044] Step 3: Based on the results obtained in step 2 Constructing pseudo snapshot data matrix in, is the virtual received signal value of the virtual array, T s is the pseudo sampling interval, the pseudo snapshot number J w = J / 2, is the pseudo signal matrix;

[0045] Step 4: Construct a uniform linear array set in Representing a collection The maximum value of , using interpolation, the sparse virtual array obtained in step 3 is transformed into a virtual uniform linear array, that is, the interpolated virtual array, and the corresponding data matrix is ​​expressed as in Representing a collection Contains but the set It is not included in Among them A z A steering matrix representing the interpolated virtual array;

[0046] Step 6: Order Construct a (12+1)×(12+1) Toeplitz matrix using the data vector z(t) in, Indicated by The Toeplitz matrix formed by Indicated by The Toeplitz matrix formed by

[0047] Step 6: According to the formula Construct a spatiotemporal correlation matrix, where the superscript "H" represents the Hermitian transpose operation. represents the array steering matrix, and the kth signal steering vector is Represents S(t) and the signal obtained by the i-th array element The correlation matrix between them, where the signal diagonal matrix S(t) = diag{R s (T s ),R s (2T s ),…,R s (J w T s )};

[0048] Step 7: According to formula s k (t-τ)≈s k (t)e -jωτ , the spatiotemporal correlation matrix obtained in step 6 Further expressed as

[0049] Step 8: According to the formula Reconstruct the Toeplitz matrix,

[0050] Step 9: According to the formula get Where J represents a switching matrix, which has 1 on the anti-diagonal and 0 elsewhere;

[0051] Step 10: According to the formula Pair Matrix Eigenvalue decomposition, where e i represents the eigenvector corresponding to the eigenvalue,

[0052] Step 11: Take The three largest eigenvalues ​​among the eigenvalues ​​constitute the signal subspace and are divided into U X and U Y Two parts;

[0053] Step 12: Construct the matrix And calculate the eigenvalue decomposition;

[0054] Step 13: Decompose U into 3×3 sub-matrices

[0055] Step 14: According to the formula Calculate eigenvalues

[0056] Step 15: According to the formula Compute the coherent signal angle estimate.

[0057] The patent method based on the above conditions is simulated by MATLAB simulation software, and the angle estimation result of the patent method is obtained as follows Figure 4 As shown, when the input azimuth angle is (0°, 21.5°, 50°), a total of 100 experiments are conducted. The experimental results show that the patented method can accurately locate all signal sources.

[0058] The second embodiment: The relationship curve between the root mean square error and the signal-to-noise ratio of the signal processing method of the patent is given. At the same time, a comparison chart of the performance analysis of other methods (coprime array interpolation method (CSCAI), coprime array interpolation method based on atomic norm (CCANM), and the patent method (CVTS-MTOEP)) is also given. The effect diagram obtained is as follows Figure 5 The application conditions of the method of the present invention are as follows:

[0059] The array model of the coprime array used in the present invention is as follows Figure 1 As shown. The coprime array consists of subarray 1 and subarray 2. Subarray 1 has 3 elements, i.e., M=3, and the element spacing is Nd=5d. Subarray 2 has 5 elements, i.e., N=5, and the element spacing is Md=3d, where d=0.5m. Except for the first element, the other elements of the coprime array will not overlap with each other. The total number of elements is L=M+N-1=7. The number of coherent signals is K=3, and the coherence coefficient (1,e jπ / 6 ,e jπ / 8 ), the arrival angle is (0°, 21.5°, 50°), the number of snapshots is 200, the signal-to-noise ratio changes from -15dB, and each step increases from 5dB to 10dB. 500 independent Monte Carlo experiments are performed, and the simulation analysis is performed using MATLAB simulation software. The performance analysis comparison curve is as follows Figure 5 shown.

[0060] Depend on Figure 5 It can be seen that the root mean square errors of the three methods all decrease with the increase of the signal-to-noise ratio. Among them, the root mean square error of the CVTS-MTOEP method of the present invention is smaller under low signal-to-noise ratio conditions, which is consistent with the assumption in the technical solution.

[0061] The third embodiment: The relationship curve between the success probability and the signal-to-noise ratio of the signal processing method of the patent is given. At the same time, a comparison chart of the performance analysis of other methods (coprime array interpolation method (CSCAI), coprime array interpolation method based on atomic norm (CCANM), and the patent method (CVTS-MTOEP)) is also given. The effect diagram obtained is as follows Figure 6 The application conditions of the method of the present invention are as follows:

[0062] The array model of the coprime array used in the present invention is as follows Figure 1 As shown. The coprime array consists of subarray 1 and subarray 2. Subarray 1 has 3 elements, i.e., M=3, and the element spacing is Nd=5d. Subarray 2 has 5 elements, i.e., N=5, and the element spacing is Md=3d, where d=0.5m. Except for the first element, the other elements of the coprime array will not overlap with each other. The total number of elements is L=M+N-1=7. The number of coherent signals is K=3, and the coherence coefficient (1,e jπ / 6 ,e jπ / 8 ), the arrival angle is (0°, 21.5°, 50°), the number of snapshots is 200, the signal-to-noise ratio changes from -15dB, and each step increases from 5dB to 10dB. 500 independent Monte Carlo experiments are performed, and the simulation analysis is performed using MATLAB simulation software. The performance analysis comparison curves are as follows: Figure 6 shown.

[0063] Depend on Figure 6 It can be seen that the success probability of the three methods increases with the increase of the signal-to-noise ratio. Among them, the success probability of the CVTS-MTOEP method of the present invention is always the highest.

[0064] Example 4: The relationship curve between the root mean square error and the number of snapshots of the signal processing method of the patent is shown. At the same time, a comparison chart of the performance analysis of other methods (coprime array interpolation method (CSCAI), coprime array interpolation method based on atomic norm (CCANM), and the patent method (CVTS-MTOEP)) is also given. The effect diagram obtained is as follows Figure 7 The application conditions of the method of the present invention are as follows:

[0065] The array model of the coprime array used in the present invention is as follows Figure 1 As shown. The coprime array consists of subarray 1 and subarray 2. Subarray 1 has 3 elements, i.e., M=3, and the element spacing is Nd=5d. Subarray 2 has 5 elements, i.e., N=5, and the element spacing is Md=3d, where d=0.5m. Except for the first element, the other elements of the coprime array will not overlap with each other. The total number of elements is L=M+N-1=7. The number of coherent signals is K=3, and the coherence coefficient (1,e jπ / 6 ,e jπ / 8), the arrival angle is (0°, 21.5°, 50°), the signal-to-noise ratio is -5dB, the number of snapshots changes from 40, and the step size increases from 20 to 240 each time. 500 independent Monte Carlo experiments are performed, and the simulation analysis is performed using MATLAB simulation software. The performance analysis comparison curve is as follows Figure 7 shown.

[0066] Depend on Figure 7 It can be seen that the root mean square errors of the three methods all decrease with the increase of the number of snapshots. Among them, the root mean square error of the CVTS-MTOEP method of the present invention always outperforms other methods when the number of snapshots is higher than 200.

[0067] The fifth embodiment: The relationship curve between the success probability and the signal-to-noise ratio of the signal processing method of the patent is given. At the same time, a comparison chart of the performance analysis of other methods (coprime array interpolation method (CSCAI), coprime array interpolation method based on atomic norm (CCANM), and the patent method (CVTS-MTOEP)) is also given. The effect diagram obtained is as follows Figure 8 The application conditions of the method of the present invention are as follows:

[0068] The array model of the coprime array used in the present invention is as follows Figure 1 As shown. The coprime array consists of subarray 1 and subarray 2. Subarray 1 has 3 elements, i.e., M=3, and the element spacing is Nd=5d. Subarray 2 has 5 elements, i.e., N=5, and the element spacing is Md=3d, where d=0.5m. Except for the first element, the other elements of the coprime array will not overlap with each other. The total number of elements is L=M+N-1=7. The number of coherent signals is K=3, and the coherence coefficient (1,e jπ / 6 ,e jπ / 8 ), the arrival angle is (0°, 21.5°, 50°), the signal-to-noise ratio is -5dB, the number of snapshots changes from 40, and the step size increases from 20 to 240 each time. 500 independent Monte Carlo experiments are performed, and the simulation analysis is performed using MATLAB simulation software. The performance analysis comparison curves are as follows Figure 8 shown.

[0069] Depend on Figure 8 It can be seen that the success probability of the three methods increases with the increase of the number of snapshots. The success probability is consistent with the rule of the root mean square error.

[0070] The specific examples described herein are merely illustrative of the present invention. A person skilled in the art may make various modifications, supplements or substitute similar methods to the specific examples described, but this will not deviate from the present invention or exceed the scope defined by the appended claims.

Claims

1. A coherent signal direction of arrival estimation method based on coprime array matrix reconstruction, wherein subarray 1 of the coprime array has M array elements, and the array element spacing is Nd, and subarray 2 has N array elements, and the array element spacing is Md; M and N are coprime integers, and M<N, d=λ / 2, λ is half wavelength, the total number of array elements L=M+N-1, and the coprime array element position set is expressed as There are K coherent signals from different angles θ=[θ1,θ2,…,θ K ] is incident on the coprime array, the number of snapshots is J, the first signal s1(t) is used as the reference signal, and the kth coherent signal is in, β k represents the signal amplitude, Δψ k =ωt represents s k The phase difference between s1(t) and s1(t), ω represents the signal frequency, and the received signal of the pth array element is expressed as Among them, n p (t) represents additive noise, l p Representing a collection The pth element of , l1 = 0, the coprime array received signal is expressed as in, represents the coherent signal vector, A represents the array manifold matrix, l p ,p=1,…,M+N-1 represents the array element position, where the superscript "T" represents the transposition operation, n(t) represents the additive Gaussian white noise, and the power is And it is not related to the signal, and is characterized in that: the method for estimating the direction of arrival includes the following steps: Step 1: The cross-correlation function of the received signal of the first array element and the received signal of the pth array element with respect to the time length τ (τ≠0) is expressed as in, Further expressed as Constructing vectors We get R(τ) = AR s (τ), according to R s (τ)=(R s (-τ) * , we get (R(-τ)) * =A * R s (τ), let the matrix A (2) and vector R (2) (τ) represents the 2nd to M+N-1th rows of the matrix A and the vector R(τ), respectively. (2) (τ) = A (2) R s (τ); Step 2: According to the (R(-τ)) obtained in step 1 * and R (2) (τ), construct the matrix in, is the received signal value of a virtual array, is the direction matrix of the virtual array, is a single snapshot signal vector composed of K virtual incident signals, where the kth virtual incident signal is expressed as k=1,2,…,K, virtual array position set express The mirror symmetric set about the origin, the total number of array elements Step 3: Based on the results obtained in step 2 Constructing pseudo snapshot data matrix in, is the virtual received signal value of the virtual array, T s is the pseudo sampling interval, the pseudo snapshot number J w =J / 2, is the pseudo signal matrix; Step 4: Construct a uniform linear array set in Representing a collection The maximum value of , using interpolation, the sparse virtual array obtained in step 3 is transformed into a virtual uniform linear array, that is, the interpolated virtual array, and the corresponding data matrix is ​​expressed as in Representing a collection Contains but the set It is not included in Among them A z A steering matrix representing the interpolated virtual array; Step 5: Order Use z(t) from step 4 to construct a (Z+1)×(Z+1) Toeplitz matrix in, Indicated by The Toeplitz matrix formed by Indicated by The Toeplitz matrix formed; Step 6: According to the formula Construct a spatiotemporal correlation matrix, where the superscript "H" represents the Hermitian transpose operation. represents the array steering matrix, and the kth signal steering vector is Represents S(t) and the signal g obtained by the i-th array element i The correlation matrix between (t), where the signal diagonal matrix S(t) = diag{R s (T s ),R s (2T s ),…,R s (J w T s )}; Step 7: According to formula s k (t-τ)≈s k (t)e -jωτ , the spatiotemporal correlation matrix obtained in step 6 Further expressed as Step 8: According to the formula Reconstruct the Toeplitz matrix, Step 9: According to the formula get Where J represents a switching matrix, which has 1 on the anti-diagonal and 0 elsewhere; Step 10: According to the formula Pair Matrix Eigenvalue decomposition, where e i represents the eigenvector corresponding to the eigenvalue, Step 11: Take The K largest eigenvalues ​​among the eigenvalues ​​constitute the signal subspace and are divided into U X and U Y Two parts; Step 12: Construct the matrix And calculate the eigenvalue decomposition; Step 13: Decompose U into K×K sub-matrices Step 14: According to the formula Calculate eigenvalues Step 15: According to the formula Compute the coherent signal angle estimate.

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