A method for direction of arrival estimation based on SAMP improved by nested array

Abstract

This invention relates to the technical field of adaptive array signal processing, and particularly to a direction-of-arrival (DOA) estimation method based on an improved SAMP (Single Amplifier Array) using nested arrays. This method addresses the technical problems in the prior art. It involves incident a far-field narrowband signal onto an extended nested array. Based on the signal's position information, the array manifold matrix is ​​obtained, leading to the array received data matrix of the nested interpolated uniform array. Then, based on the characteristics of the compressed sensing matrix, the array manifold is extended to obtain a DOA estimation mathematical model. This model is input into an improved adaptive matched-tracking algorithm to reconstruct the original input signal. Finally, the MUSIC algorithm is used for spectral peak search. This method can accurately achieve DOA estimation under conditions of small snapshots, low signal-to-noise ratio, and source coherence; it has low computational complexity; it can accurately recover the source with relatively few samples; and it effectively improves the DOA estimation accuracy and spatial resolution of nested arrays.

Description

Technical Field

[0001] This invention relates to the technical field of adaptive array signal processing, and more particularly to a direction-of-arrival estimation method based on a nested array-based improved SAMP. Background Technology

[0002] Subspace-based methods rely on the orthogonality of the signal and noise subspaces. However, when the number of snapshots decreases or the signal-to-noise ratio drops, the signal and noise subspaces become non-orthogonal, severely degrading the estimation performance of these methods. Furthermore, traditional subspace-based algorithms cannot distinguish coherent signals. For coherent signals, there are spatial smoothing (SSM) algorithms, decoherence algorithms based on higher-order cumulants, and subspace fitting algorithms. However, spatial smoothing algorithms not only cause a loss of antenna array aperture but also have a narrow applicability, only suitable for uniform linear and uniform planar arrays. Subspace fitting algorithms require multi-dimensional searches during operation, resulting in unacceptable computational complexity and making them impractical for real-world applications. Summary of the Invention

[0003] To overcome the shortcomings of existing traditional subspace algorithms in estimating direction of arrival (DOA) with low accuracy or even failure under conditions such as small snapshots, low signal-to-noise ratio, and coherent source information, this invention provides a DOA estimation method based on a nested array-based improved SAMP.

[0004] This invention provides a direction-of-arrival estimation method based on a nested array-based improved SAMP, comprising the following steps:

[0005] Step 1: Assume there is A far-field narrowband signal is incident on the second order On a nested array, second order The nested array is composed of two nested uniform subarrays, uniform subarray one and uniform subarray two. The number of elements in uniform subarray one is... The spacing between array elements is The number of elements in a uniform subarray is The spacing between array elements is ,and ,in ; by second order The nested array yields the set of element positions of the nested array. ,in ;pass Obtain the array manifold vector of the nested array ,in ;Depend on Obtain the array manifold matrix ,in Therefore, the received signal of the nested array at time t is represented as: ,in For the received signal vector, The source signal vector, This indicates that the mean is 0 and the energy is... Additive white Gaussian noise; the covariance matrix of the received signal is ,in ;

[0006] Step 2: Extend the second-order M-element nested array: The array degrees of freedom of the optimal second-order nested array are... ,in The second-order statistical information of a second-order nested array is generated by performing a matrix-vector transformation. Each virtual array element has a range of [number] elements. For second order If the nested array is expanded such that the spacing between adjacent elements in the uniform subarray 2 is equal to the spacing between elements in the uniform subarray 1, then the updated set of element positions can be determined as follows: ,in ; From the set of array element positions The array manifold vector of the expanded nested array is: ,in ]; and thus the array manifold matrix of the extended nested array is obtained as ,in Then the received signal of the virtual array at time t is: ,in The covariance matrix of the received signal of the virtual array is: ;in It is the covariance matrix of the incident signal, E is the expectation operator, and the superscript H indicates the conjugate transpose;

[0007] Step 3: Based on the spatial domain partitioning method, divide the expanded nested array into array manifold matrices. from Dimensional extension Dimension, where N >> K, yields the overcomplete array manifold matrix. ,in The mathematical model for estimating the direction of arrival (DOA) of a sparse signal is: ,in It is the measurement matrix in compressed sensing. For sparse signals, It is additive white Gaussian noise; and because the signal is sparse. It is sparse throughout the entire airspace, so It is also a perception matrix;

[0008] Step 4: Dimensional measurement rectangle , Dimensional observations After inputting a sparse adaptive matching pursuit algorithm with a step size of s, the output is... sparse signals The improved sparsity adaptive matching pursuit algorithm includes the following sub-steps:

[0009] S4.1 Parameter initialization, i.e., setting the residual Support set Sparsity L=s, number of iterations ;

[0010] S4.2 Calculation ,choose In The maximum value, The overcomplete array manifold matrix corresponding to each maximum value Serial number Form a set ;

[0011] S4.3 Regularization processing, in sets Finding a subset ,satisfy ; Select all subsets that meet the requirements. The one with the most energy ;

[0012] S4.4, Update the index set and atomic sets ,in , ;

[0013] S4.5. Calculate S using the least squares method, i.e. ;

[0014] S4.6, Update Residuals From step S4.5 Select the one with the largest absolute value Item is recorded as Corresponding Column L in the middle is denoted as , denote the set ;

[0015] S4.7, if residual If the residual is 0, stop the iteration and proceed to step S4.8; if the residual is 0, proceed to step S4.8. Then update the sparsity. And return to step S4.2 to continue the iteration, where ;if If the iteration stops, proceed to step S4.8; otherwise, return to step S4.2 to continue the iteration.

[0016] S4.8, Based on the perception matrix The spatial sparsity scheme of the middle column will be iteratively obtained sparse signals ;

[0017] Step 5: Use the MUSIC algorithm to search for spectral peaks and obtain the direction of arrival: Based on the improved sparsity adaptive matching pursuit algorithm in Step 4, obtain the reconstructed direction of arrival. sparse signals ,calculate sparse signals The covariance matrix is ​​obtained by performing eigenvalue decomposition on the covariance matrix to obtain the signal subspace and noise subspace. The spatial spectrum function of the signal subspace and noise subspace is calculated by using the orthogonality of the signal subspace and noise subspace. The angle corresponding to the maximum value of the spatial spectrum function is the direction of arrival of the far-field narrowband signal.

[0018] Compared with the prior art, the technical solution provided by this invention has the following advantages: it can accurately achieve direction of arrival estimation under conditions such as small snapshots, low signal-to-noise ratio, and source coherence; it has low computational complexity; it can accurately recover the source by sampling less data; it effectively improves the DOA estimation accuracy and spatial resolution of nested arrays; and the invention has verified its good performance through simulation experiments, which has important theoretical and engineering value for spatial spectrum estimation. Attached Figure Description

[0019] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with the invention and, together with the description, serve to explain the principles of the invention.

[0020] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, for those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0021] Figure 1 This is a two-level nested structure diagram.

[0022] Figure 2 The number of array elements is A schematic diagram of the physical array arrangement.

[0023] Figure 3 This is a schematic diagram of the array structure of a 9-element second-order nested array.

[0024] Figure 4 Spatial spectra of four algorithms under the conditions of 9 input array elements, 3 information sources, and incoherent information sources.

[0025] Figure 5 The spatial spectra of five algorithms are given under the conditions of 9 input array elements, 3 information sources, and coherent information sources.

[0026] Figure 6 The graph shows the minimum mean square error (RMSE) of the four algorithms as a function of the signal-to-noise ratio when the input is an incoherent source.

[0027] Figure 7 The graph shows the minimum mean square error (RMSE) of the four algorithms as a function of the signal-to-noise ratio when the input is a coherent source.

[0028] Figure 8 The graph shows the resolution success rate of the four algorithms as a function of signal-to-noise ratio (SNR) under different SNR conditions. Detailed Implementation

[0029] To better understand the above-mentioned objectives, features, and advantages of the present invention, the solutions of the present invention will be further described below. It should be noted that, unless otherwise specified, the embodiments of the present invention and the features thereof can be combined with each other.

[0030] In this description, it should be noted that the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance. It should also be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "joint" should be interpreted broadly. For example, they can refer to fixed connections, detachable connections, or integral connections; they can refer to mechanical connections or electrical connections; they can refer to direct connections or indirect connections through an intermediate medium; and they can refer to the internal connection between two components. Those skilled in the art can understand the specific meaning of the above terms according to the specific circumstances.

[0031] Many specific details are set forth in the following description in order to provide a full understanding of the invention, but the invention may also be practiced in other ways different from those described herein; obviously, the embodiments in the specification are only some embodiments of the invention, and not all embodiments.

[0032] The following is in conjunction with the appendix Figures 1 to 8 Specific embodiments of the present invention will be described in detail below.

[0033] In one embodiment, the present invention discloses a direction-of-arrival estimation method based on a nested array-improved SAMP, comprising the following steps:

[0034] Step 1: Assume there is A far-field narrowband signal is incident on the second order On a nested array, second order The nested array is composed of two nested uniform subarrays, uniform subarray one and uniform subarray two. The number of elements in uniform subarray one is... The spacing between array elements is The number of elements in a uniform subarray is The spacing between array elements is ,and ,in ; by second order The nested array yields the set of element positions of the nested array. ,in In this specific embodiment And the specific set of array element positions of the nested array for ;pass Obtain the array manifold vector of the nested array ,in ;Depend on Obtain the array manifold matrix ,in Therefore, the received signal of the nested array at time t is represented as: ,in For the received signal vector, The source signal vector, This indicates that the mean is 0 and the energy is... Additive white Gaussian noise; the covariance matrix of the received signal is ,in ;

[0035] Step 2: Extend the second-order M-element nested array: The array degrees of freedom of the optimal second-order nested array are... ,in The second-order statistical information of a second-order nested array is generated by performing a matrix-vector transformation. Each virtual array element has a range of [number] elements. For second order If the nested array is expanded such that the spacing between adjacent elements in the uniform subarray 2 is equal to the spacing between elements in the uniform subarray 1, then the updated set of element positions can be determined as follows: ,in ; From the set of array element positions The array manifold vector of the expanded nested array is: ,in ]; and thus the array manifold matrix of the extended nested array is obtained as ,in Then the received signal of the virtual array at time t is: ,in The covariance matrix of the received signal of the virtual array is: ;in It is the covariance matrix of the incident signal, E is the expectation operator, and the superscript H indicates the conjugate transpose; the nested array is extended by matrix-vector transformation, which increases the aperture of the nested array and significantly increases the degree of freedom of the direction of arrival (DOA);

[0036] Step 3: Based on the spatial domain partitioning method, divide the expanded nested array into array manifold matrices. from Dimensional extension Dimension, where N >> K, yields the overcomplete array manifold matrix. ,in The mathematical model for estimating the direction of arrival (DOA) of a sparse signal is: ,in As a measurement matrix in compressed sensing For sparse signals, It is additive white Gaussian noise; and because the signal is sparse. It is sparse throughout the entire airspace, so It is also a perception matrix;

[0037] Step 4: Dimensional measurement rectangle , Dimensional observations And after inputting an improved sparseness adaptive matching pursuit algorithm (ISAMP) with a step size of s, the output is... sparse signals The traditional sparse adaptive matching pursuit algorithm (SAMP) achieves sparsity adaptation in stages. The specific process is as follows: First, an initial step size is set. The first stage focuses on sparsity. The signal is then reconstructed through iterative iterations. If the sparsity is appropriate, the support set size is maintained at that sparsity for iteration until the iteration termination condition is met. Otherwise, the residual after iteration will increase, indicating insufficient sparsity, and the support set needs to be increased to improve the sparsity. Continue iterating until the iteration stopping condition is met and the result is output; the improved sparsity adaptive matching pursuit algorithm in this application includes the following sub-steps:

[0038] S4.1 Parameter initialization, i.e., setting the residual Support set Sparsity L=s, number of iterations ;

[0039] S4.2 Calculation ,choose In The maximum value, The overcomplete array manifold matrix corresponding to each maximum value Serial number Form a set ;

[0040] S4.3 Regularization processing, in sets Finding a subset ,satisfy ; Select all subsets that meet the requirements. The one with the most energy ;

[0041] S4.4, Update the index set and atomic sets ,in , ;

[0042] S4.5. Calculate S using the least squares method, i.e. ;

[0043] S4.6, Update Residuals From step S4.5 Select the one with the largest absolute value Item is recorded as Corresponding Column L in the middle is denoted as , denote the set ;

[0044] S4.7, if residual If the residual is 0, stop the iteration and proceed to step S4.8; if the residual is 0, proceed to step S4.8. Then update the sparsity. And return to step S4.2 to continue the iteration, where ;if If the iteration stops, proceed to step S4.8; otherwise, return to step S4.2 to continue the iteration.

[0045] S4.8, Based on the perception matrix The spatial sparsity scheme of the middle column will be iteratively obtained sparse signals ;

[0046] Step 5: Use the MUSIC algorithm to search for spectral peaks and obtain the direction of arrival: Based on the improved sparsity adaptive matching pursuit algorithm in Step 4, obtain the reconstructed direction of arrival. sparse signals ,calculate sparse signals The covariance matrix is ​​obtained by performing eigenvalue decomposition on the covariance matrix to obtain the signal subspace and noise subspace. The spatial spectrum function of the signal subspace and noise subspace is calculated by using the orthogonality of the signal subspace and noise subspace. The angle corresponding to the maximum value of the spatial spectrum function is the direction of arrival of the far-field narrowband signal.

[0047] In the proposed method for direction-of-arrival (DOA) estimation based on an improved SAMP using nested arrays, a far-field narrowband signal is incident on an expanded nested array. Based on its position information, the array manifold matrix is ​​obtained, leading to the array received data matrix of the nested interpolated uniform array. Then, based on the characteristics of the compressed sensing matrix, the array manifold is expanded to obtain a DOA estimation mathematical model. This model is input into an improved adaptive matched pursuit algorithm to reconstruct the original input signal. Finally, the MUSIC algorithm is used for spectral peak search. The improved sparse adaptive matched pursuit algorithm in step four adds regularization processing to the SAMP algorithm, reselecting the initially chosen atoms for more accurate source reconstruction. Furthermore, by using the DOA estimation mathematical model from step three as input to the improved SAMP algorithm in step four, the signal is accurately reconstructed with fewer samples, effectively reducing computational complexity.

[0048] Figure 2 The number of array elements is A schematic diagram of the physical array arrangement, wherein, Figure 2 The first line indicates the actual arrangement of the physical array; Figure 2 The second line indicates the arrangement of the virtual array after the nested array has been expanded. Figure 3 This is a schematic diagram of a physical array with 9 array elements.

[0049] Figure 4 It is the spatial spectrum when the incident signal is incoherent, with 128 snapshots, an incident angle of [-20°, 0°, 30°], and a signal-to-noise ratio of 0dB. Figure 4 In this context, SAMP stands for Sparse Adaptive Matching Pursuit Algorithm, NA-SAMP is a Nested Array-Sparse Adaptive Matching Pursuit Algorithm, ISAMP is an Improved Sparse Adaptive Matching Pursuit Algorithm, and NA-ISAMP is a Nested Array-Improved Sparse Adaptive Matching Pursuit Algorithm. Figure 4 It can be seen that when the incident signal is incoherent, all four algorithms can estimate the incident signal with relatively high accuracy. Simulation results show that the NA-ISAMP algorithm proposed in this invention has sharper spectral peaks, i.e., better estimation performance, because it utilizes the scalability of nested arrays to expand the virtual aperture of the array, thereby greatly improving the estimation accuracy.

[0050] Figure 5 This is the spatial spectrum when the incident signal is coherent, with 128 snapshots, an incident angle of [-20°, 0°, 30°], and a signal-to-noise ratio of 0dB. Figure 5 It can be seen that when the incident signal is a coherent signal, the traditional MUSIC algorithm cannot effectively distinguish the signal source, while the other four algorithms can estimate the incident signal with relatively high accuracy. It can be observed that the traditional MUSIC algorithm has a small spatial spectral peak value, making it ineffective in distinguishing the signal source. For example... Figure 5 As shown, the spatial spectrum of the improved sparsity adaptive matching pursuit algorithm (NA-ISAMP) under nested arrays is the sharpest, and its estimation effect is the best. This is because the nested array is expanded, increasing the array aperture and improving the array's degrees of freedom, resulting in more accurate DOA estimation. Simultaneously, the ISAMP algorithm solves the rank deficiency problem that occurs during the virtual expansion of the nested array and overcomes the rank loss drawback of using spatial smoothing algorithms when dealing with coherent input signals.

[0051] Figure 6 This is a graph showing the root mean square error as a function of the signal-to-noise ratio when the input signal is incoherent. Figure 7 This is a graph showing the change of root mean square error as a function of signal-to-noise ratio when the input signal is a coherent signal. Figure 6 and Figure 7 The corresponding number of Monte Carlo independent trials was set to 500, with the signal-to-noise ratio varying from -20 dB to 5 dB. The root mean square error (RMSE) in the experimental simulation was defined as:

[0052] ,

[0053] In the above formula, This represents the number of information sources, with N=500 independent experiments. θ is the angle estimate of the nth experiment from the kth source. n It is the true angle of arrival of the k-th source.

[0054] from Figure 6 and Figure 7 It can be seen that as the signal-to-noise ratio (SNR) increases, the root mean square (RMSE) errors of the SAMP, NA-SAMP, ISAMP, and NA-ISAMP algorithms gradually decrease. Furthermore, it can be observed that at low SNR, the NA-ISAMP algorithm proposed in this invention has a significantly lower error than the other three algorithms. When the SNR is very low, the RMSE value is very small, meaning that this invention can still accurately estimate the direction of arrival (ROA) of the signal.

[0055] Figure 8This represents the resolution success rate of four algorithms under different signal-to-noise ratios (SNRs). The number of independent Monte Carlo trials was set to 500, and the SNR varied from -20 dB to 20 dB. The resolution success rate in the experimental simulation was defined as follows:

[0056] ,

[0057] S k This represents the number of times the k-th source was successfully estimated in N DOA estimation experiments.

[0058] It can be observed that the resolution success rate gradually increases with the increase of the signal-to-noise ratio (SNR). When the SNR is low, the POR of the NA-ISAMP algorithm proposed in this invention is significantly higher than that of the other three algorithms. When the SNR is 2.5 dB, the resolution success rate reaches 100%.

[0059] The above description is merely a specific embodiment of the present invention, enabling those skilled in the art to understand or implement the present invention. Although detailed descriptions have been provided with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments, and they should all be covered within the protection scope of the claims.

Claims

1. A direction-of-arrival estimation method based on a nested array-based improved SAMP, characterized in that, Includes the following steps: Step 1: Assume there is A far-field narrowband signal is incident on the second order On a nested array, second order The nested array is composed of two nested uniform subarrays, uniform subarray one and uniform subarray two. The number of elements in uniform subarray one is... The spacing between array elements is The number of elements in a uniform subarray is The spacing between array elements is ,and ,in ; by second order The nested array yields the set of element positions of the nested array. ,in ;pass Obtain the array manifold vector of the nested array ,in ;Depend on Obtain the array manifold matrix ,Right now Therefore, the received signal of the nested array at time t is represented as: ,in For the received signal vector, The source signal vector, This indicates that the mean is 0 and the energy is... Additive white Gaussian noise; the covariance matrix of the received signal is ,in ; Step 2: Extend the second-order M-element nested array: The array degrees of freedom of the optimal second-order nested array are... ,in The second-order statistical information of a second-order nested array is generated by performing a matrix-vector transformation. Each virtual array element has a range of [number] elements. For second order If the nested array is expanded such that the spacing between adjacent elements in the uniform subarray 2 is equal to the spacing between elements in the uniform subarray 1, then the updated set of element positions can be determined as follows: ,in ; From the set of array element positions The array manifold vector of the expanded nested array is: ,in ]; and thus the array manifold matrix of the extended nested array is obtained as ,in Then the received signal of the virtual array at time t is: ,in The covariance matrix of the received signal of the virtual array is: ;in It is the covariance matrix of the incident signal, E is the expectation operator, and the superscript H indicates the conjugate transpose; Step 3: Based on the spatial domain partitioning method, divide the expanded nested array into array manifold matrices. from Dimensional extension Dimension, where N >> K, yields the overcomplete array manifold matrix. ,in The mathematical model for estimating the direction of arrival (DOA) of a sparse signal is then: ,in It is the measurement matrix in compressed sensing. For sparse signals, It is additive white Gaussian noise; and because the signal is sparse. It is sparse throughout the entire airspace, so It is also a perception matrix; Step 4: Dimensional measurement rectangle , Dimensional observations After inputting a sparse adaptive matching pursuit algorithm with a step size of s, the output is... sparse signals The improved sparsity adaptive matching pursuit algorithm includes the following sub-steps: S4.1 Parameter initialization, i.e., setting the residual Support set Sparsity L = s iteration number ; S4.2 Calculation ,choose In The maximum value, The overcomplete array manifold matrix corresponding to each maximum value Serial number Form a set ; S4.3 Regularization processing, in sets Finding a subset ,satisfy ; Select all subsets that meet the requirements. The one with the most energy ; S4.4, Update the index set and atomic sets ,in , ; S4.

5. Calculate S using the least squares method, i.e. ; S4.6, Update Residuals From step S4.5 Select the one with the largest absolute value Item is denoted as Corresponding Column L in the middle is denoted as , denote the set ; S4.7, if residual If the residual is 0, stop the iteration and proceed to step S4.8; if the residual is 0, proceed to step S4.

8. Then update the sparsity. And return to step S4.2 to continue the iteration, where ;if If the iteration stops, proceed to step S4.8; otherwise, return to step S4.2 to continue the iteration. S4.8, Based on the perception matrix The spatial sparsity scheme of the middle column will be iteratively obtained sparse signals ; Step 5: Use the MUSIC algorithm to search for spectral peaks and obtain the direction of arrival: Based on the improved sparsity adaptive matching pursuit algorithm in Step 4, obtain the reconstructed direction of arrival. sparse signals ,calculate sparse signals The covariance matrix is ​​obtained by performing eigenvalue decomposition on the covariance matrix to obtain the signal subspace and noise subspace. The spatial spectrum function of the signal subspace and noise subspace is calculated by using the orthogonality of the signal subspace and noise subspace. The angle corresponding to the maximum value of the spatial spectrum function is the direction of arrival of the far-field narrowband signal.

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