A DOA estimation method based on multi-frequency nested MIMO array

Abstract

The application discloses a DOA estimation method based on a multi-frequency nested MIMO array, and is based on a traditional nested MIMO array and a low-rank matrix completion theory. First, a set of mutually orthogonal multi-frequency detection waves are sent by a transmitting array of the nested MIMO array, are reflected by a target, are received by a receiving array of the nested MIMO array, are subjected to matched filtering, and then a plurality of groups of received signals are generated. A summation cooperative array with uniformly distributed holes is constructed. Signal reconstruction and low-rank matrix completion operations are performed on the summation cooperative array signals, and finally, DOA estimation is performed. Compared with a traditional DOA estimation method based on a sparse array, the DOA estimation method can improve the maximum estimable incident target number and resolution of the array, and reduce complexity, cost and size. Compared with other DOA estimation methods based on low-rank matrix completion and sparse arrays, the application optimizes the array structure for low-rank matrix completion, and improves the array aperture expansion efficiency.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of array signal processing, and particularly relates to a DOA estimation method based on a multi-frequency nested MIMO array. BACKGROUND

[0002] Direction of arrival (DOA) estimation is one of the important directions of array signal processing, and is widely applied in radar, communication and other fields. Many classical DOA estimation algorithms have been proposed in the academic field, and the subspace-based algorithm is the most popular, such as MUSIC and ESPRIT algorithms. Both in estimation accuracy and in computational complexity, the subspace-based algorithm has certain advantages over other algorithms. In actual engineering applications, a linear uniform array, a circular array or a planar array with a spacing d (half wavelength) between array elements is usually used to receive incoming signals. With the development of technology, the requirements for the degrees of freedom and resolution of DOA estimation are further improved. The intuitive method to achieve this goal is to expand the aperture of the array, i.e. to increase the number of array elements. Of course, increasing the number of physical array elements can effectively improve the degrees of freedom and resolution of DOA estimation, but the cost problem cannot be ignored. At the same time, with the increase of the working frequency of the array, the distribution of the array becomes more and more dense, which may cause problems such as mutual coupling. Moreover, in the underwater detection scene (low-frequency application), a too large array size may also cause deployment problems. In view of the above problems, the academic field has proposed sparse arrays such as coprime arrays and nested arrays. These methods construct a pseudo-difference co-array, but these methods use multi-ping data to expand the array aperture, so the time-invariant characteristics of the signal are required to be high, and the array size cannot be reduced. The characteristics of the MIMO array can also be used to expand the array aperture, and this method can reduce the array size and does not consume the number of pings. In order to further improve the performance of the array, the combination of MIMO array and special array structures such as nested coprime array can improve the potential of array design and bring more rich array design schemes. It is worth mentioning that in the array design, the virtual array formed at last may still be a sparse array, and the common choice is to take the largest continuous array for DOA estimation. In order to better utilize the information of all virtual arrays, a direct idea is to complete the missing information, and the matrix low-rank completion is introduced into the DOA estimation because it can well restore the low-rank characteristics of the second-order moment of the array signal.

[0003] Most of the current matrix low-rank completion algorithms are directly applied to existing sparse arrays, and few improvements are made on the structure of the array. Therefore, it is urgent to propose a DOA estimation method based on a multi-frequency nested MIMO array. SUMMARY

[0004] The purpose of the present application is to solve the above-mentioned defects in the prior art, and to provide a DOA estimation method based on a multi-frequency nested MIMO array, which meets the requirements of a low-rank completion algorithm for a matrix. The method comprises the following steps: a multi-frequency orthogonal probe wave is transmitted by a transmitting array of a nested MIMO array, and the probe wave is reflected by a target and then received by a receiving array of the nested MIMO array; a plurality of sets of received signal vectors x(f, t) are generated after matching filtering; a summing cooperative array with a relatively uniform distribution of holes is constructed; and a virtual array signal is constructed by setting the holes to zero. The second moment of the virtual array signal is calculated, and the DOA is estimated after matrix completion of the second moment. Compared with other methods, the method can improve the number of estimable incident targets and the resolution of the array, and reduce the complexity, cost and size of the DOA estimation method.

[0005] The purpose of the present application can be achieved by adopting the following technical solutions:

[0006] A DOA estimation method based on a multi-frequency nested MIMO array, the method comprising the following steps:

[0007] S1. According to the wave speed c and the frequency f0 of the propagation medium, the wavelength λ is calculated as λ=c / f0, and the minimum unit of the element spacing of the nested MIMO array is set as half the wavelength, i.e. d=0.5λ=0.5c / f0;

[0008] S2. The number of elements M of the receiving array and the number of elements N of the transmitting array of the nested MIMO array are determined;

[0009] S3. The transmitting array of the nested MIMO array transmits a multi-frequency orthogonal probe wave with a frequency of f z ∈{2f0,3f0,(3MN+1)f0}, and the receiving array of the nested MIMO array receives the probe wave echo reflected by the target and performs matching filtering to generate a plurality of sets of received signal vectors x(f z , t):

[0010] x(f z , t)={x(2f0,t),x(3f0,t),x((3MN+1)f0,t)}

[0011] wherein x(2f0,t), x(3f0,t), and x((3MN+1)f0,t) are the first, second, and third received signal components, respectively;

[0012] S4. The received x(f z , t) is reconstructed into a virtual array signal X(t);

[0013] S5. The second moment R XX of the virtual array signal X(t) is calculated;

[0014] S6, the low-rank matrix completion is performed on the second moment RXX to obtain

[0015] S7, DOA estimation is performed on .

[0016] Further, in step S2, the number of receiving array elements M of the nested MIMO array and the number of transmitting array elements N are nested, wherein the set of transmitting array positions is: The set of receiving array positions is: Through such a design, the nested structure can be realized, and the nested MIMO array can expand the equivalent summation cooperative array with equidistant distribution and

[0017] Further, in step S3, the set of summation cooperative array positions corresponding to the first receiving signal component x(2f0, t) is as follows:

[0018] The set of summation cooperative array positions corresponding to the second receiving signal component x(3f0, t) is as follows:

[0019]

[0020] The set of summation cooperative array positions corresponding to the third receiving signal component x((3MN+1)f0, t) is as follows:

[0021] The set of summation cooperative array values is wherein and constitute a classic nested array structure, and constitute a classic coprime array structure, and by using these characteristics, the summation cooperative array with approximately uniformly distributed holes can be constructed

[0022] Further, in step S4, the virtual array with element spacing d is assumed: The virtual array corresponding to the signal X(t) is composed of x(f z , t) and the partial hole elements, wherein the hole elements are elements with a value of O, and the composition relationship is:

[0023]

[0024] wherein X k (t) refers to the virtual array corresponding to the signal vector X(t) The signal received by the element with middle position kd, x k (f z , t) refers to the signal vector x(f z , t) corresponding to the virtual array The signal received by the element with middle position kd, x k (t) has three cases:

[0025] If the element with middle position kd corresponds to a single signal of x(f z , t), i.e. (f z / f0)(n+Nm)=k has only one integer solution, X k (t)=x k (f z , t);

[0026] If the element with middle position kd corresponds to multiple signals of x(f z , t), i.e. (f z / f0)(n+Nm)=k has two integer solutions, X k (t)=x k (2f0, t);

[0027] If the element with middle position kd has no corresponding signal in x(f z , t), i.e. (f z / f0)(n+Nm)=k has no integer solution, i.e. refers to and When X k (t) is directly set to zero.

[0028] The above process is to construct the corresponding virtual array signal X(t), because is a uniform linear array, and the second moment of the array signal corresponding to the ideal uniform linear array has the properties of Toeplitz and Hermitian matrix. Toeplitz matrix: the elements on the main diagonal of the Toeplitz matrix are equal, and the elements on the lines parallel to the main diagonal are also equal, that is, the Toeplitz matrix is determined by its first row and first column. Hermitian matrix refers to a self-conjugate matrix. Each element in the ith row and jth column of the matrix is equal to the conjugate of the element in the jth row and ith column. If a matrix has these two properties, it means that the matrix can be determined by the first row or first column of the matrix. These matrix properties can be used as prior information to participate in the matrix completion in the subsequent steps.

[0029] Further, in the step S5, the second moment R XX is a sparse matrix. Because the virtual array signal X(t) is composed of x(f z , t) and the hole elements, the second moment of X(t) will find that there are a large number of rows and columns of all zeros in R XX . Such R XX cannot be directly applied to the DOA estimation. It is necessary to cooperate with the prior information of the Toeplitz matrix and the Hermitian matrix in the subsequent steps to complete the matrix.

[0030] Further, in the step S6, because the second moment R XX is a sparse matrix, it cannot be directly used for DOA estimation, so it is necessary to complete the matrix of the second moment R XX , because R XX is generally low rank in the scene of DOA estimation. The rank of the matrix measures the correlation between the rows and columns of the matrix. If the rows or columns of the matrix are linearly independent, it is a full rank matrix. If the rank of the matrix is much smaller than its row or column number, the matrix is called low rank. In fact, a low rank matrix contains a lot of redundant information, which can be used for matrix completion by virtue of this low rank.

[0031] In addition, because of the composition of the array signal X(t), there are a large number of rows and columns of all zeros in R XX , which completely destroys the low rank of the matrix, so it cannot be directly completed. At the same time, in order to solve this problem, some prior information of R XX , namely the Toeplitz matrix and the Hermitian matrix, is needed. The convex optimization target matrix is set to a matrix with the Toeplitz matrix and the Hermitian matrix.

[0032] Specifically as follows: assuming a complex matrix with Toeplitz and Hermitian matrix properties represents the matrix with v as the first row, according to the Toeplitz matrix and the Hermitian matrix properties , it is determined that the second moment R XX is defined as an observation matrix, and the completed matrix is defined as the original matrix An observation mask matrix B with the same dimension as R XX is constructed: the positions corresponding to the missing elements in R XX are represented by 0, and the positions corresponding to the existing elements in R XX are represented by 1, and then the completed matrix is obtained from the observation matrix R XX by solving the following convex optimization problem

[0033]

[0034] ||.|| F is the Frobenious norm, represents the trace of matrix , ζ is a regularization coefficient, and the value range is 0 to 1, represents matrix is semi-positive, and ⊙ represents Hadamard product. Through the above steps, the matrix completion of R XX can be realized. The regularization coefficient ζ has a certain influence on the speed and accuracy of solving the convex optimization.

[0035] The present application has the following advantages and effects relative to the prior art:

[0036] (1) The array structure proposed in the present application uses M+N array elements to construct a virtual array with 3M 2 N 2 +MN array elements. Compared with similar schemes, the virtual array constructed by the present application has more array elements.

[0037] (2) The virtual array signal constructed by the present application is the second moment of the uniform linear array signal, so it can be applied to most DOA estimation algorithms. The virtual array signal constructed by similar schemes is a single-snapshot array signal, and can only be applied to DOA estimation algorithms that are not sensitive to the number of snapshots. Compared with the single application scenario of the virtual array signal constructed by similar schemes, the virtual array signal constructed by the present application can further construct virtual arrays suitable for different scenarios.

[0038] (3) The virtual array signal constructed by the present application has certain anti-interference ability. Even if a small part of the signal is missing due to hardware problems, the present application can still work normally. BRIEF DESCRIPTION OF DRAWINGS

[0039] The accompanying drawings, which are included to provide a further understanding of the present application and are incorporated in and constitute a part of this application, illustrate embodiments of the present application and serve to explain the present application. In the drawings:

[0040] Figure 1 is a nested MIMO array position diagram of a DOA estimation method based on a multi-frequency nested MIMO array of the present application;

[0041] Figure 2 is a signal vector processing flowchart of a DOA estimation method based on a multi-frequency nested MIMO array of the present application;

[0042] Figure 3 is a parameter schematic diagram of low-rank matrix completion of a DOA estimation method based on a multi-frequency nested MIMO array of the present application;

[0043] Figure 4A position schematic diagram of a nested MIMO array and sum cooperative array and virtual array of a DOA estimation method based on a multi-frequency nested MIMO array of the present application;

[0044] Figure 5 A schematic diagram of B in embodiment 1 of the present application, a DOA estimation method based on a multi-frequency nested MIMO array;

[0045] Figure 6 A spatial spectrum simulated in embodiment 1 of the present application, a DOA estimation method based on a multi-frequency nested MIMO array;

[0046] Figure 7 A simulation performance effect diagram in embodiment 2 of the present application, a DOA estimation method based on a multi-frequency nested MIMO array. DETAILED DESCRIPTION

[0047] In order to make the purpose, technical scheme and advantages of the embodiments of the present application clearer, the technical scheme in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are some embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor fall within the protection scope of the present application.

[0048] Embodiment 1

[0049] This embodiment mainly verifies and demonstrates whether the nested MIMO array can distinguish the target direction angle far beyond the number of array elements itself. In this embodiment, MATLAB is used for simulation experiment and demonstration of the embodiment. The step process is as follows:

[0050] S1, assuming that there are 26 uncorrelated far-field targets, assuming that the target direction angle is θ k ∈{-50, -46, -42,..., 42, 46, 50}, according to the wave speed c=1500m / s of the propagation medium and the design frequency f0=5khz, the wavelength λ=c / f0=0.3m is obtained, and the minimum unit spacing of the sensor is set to half the wavelength d=0.5λ=0.5c / f0=0.15m.

[0051] S2, determine the number of receiving array elements M=2 and the number of transmitting array elements N=2 of the nested MIMO array; determine the transmitting array position set: The receiving array position set is: The transmitting frequency of the transmitting array of the nested MIMO array is f z The multiple-frequency orthogonal detection wave of ∈{2f0, 3f0, 13f0}.

[0052] S3, The transmit array of the nested MIMO array has a transmit frequency of f. z A multi-frequency orthogonal probe wave ∈{2f0, 3f0, 13f0} is used. A nested MIMO array receiver array receives and matches the probe wave echo reflected from the target, generating multiple sets of received signal vectors x(f z ,t),

[0053] x(f z , t)={x(2f0,t),x(3f0,t),x(13f0,t)}

[0054] The reception and matched filtering process is as follows: Taking the 0th transmitting element as the reference element, due to the time delay caused by the different positions between the transmitting elements, the detected wave received by target k can be expressed as:

[0055]

[0056] Where p t，n It is the location of the nth element of the transmission array. st n (f z ,t) is the probe wave emitted by the nth element, with frequency f z For the wave ∈{2f0, 3f0, 13f0}, after reflection from the target k, the probe wave will exhibit varying degrees of amplitude attenuation and phase change:

[0057]

[0058] Where η k (f z ,t) represents the amplitude attenuation caused by reflection; γ k (f z ,t) represents the phase change caused by reflection, which is related to the frequency f. z Related to target k. The reflected signal is reflected back to the array location and received by the receiving array. Here, the 0th receiving element is taken as the reference point. These reflected signals are simplified to far-field signals. Similar to traditional DOA estimation, based on the time delay of the far-field signals arriving at different receiving elements, the expression for the signal received by the m-th receiving element can be obtained as follows:

[0059]

[0060] Where P r，m For receiving array The position of the m-th array element Where w m (f z ,t) is the noise level, assumed to conform to The random process. This invention does not consider the influence of frequency on the reflection coefficient, and the formula is expanded as follows:

[0061]

[0062] Definition The reflection coefficient s k (t) is set to a phase and amplitude obeying a Gaussian random process with mean 0 and variance 1. The above formula is matched filtered to obtain the following received signal model, i.e. the signal expression of the nth array element sending, being reflected by the target, and being received by the mth receiving array element is as follows:

[0063]

[0064] According to the signal frequency f z ∈{2f0,3f0,13f0}, and finally x m，n (t) is filtered into 3 groups of signals: wherein x(2f0,t), x(3f0,t), x(13f0,t) are respectively the first, second and third received signal components;

[0065] The position set of the summation cooperative array corresponding to the first received signal component x(2f0,t) is as follows:

[0066]

[0067] The position set of the summation cooperative array corresponding to the second received signal component x(3f0,t) is as follows:

[0068]

[0069] The position set of the summation cooperative array corresponding to the third received signal component x(13f0,t) is as follows:

[0070]

[0071] The summation cooperative array value set is

[0072] The above obtains the position parameters of the summation cooperative array

[0073]

[0074] According to the above process to realize the simulation process, assuming that the matching filter effect is ideal, according to and The array signal of the simulation is constructed. According to The constructed signal can be expressed by the following formula:

[0075]

[0076] According to The constructed signal can be expressed by the following formula:

[0077]

[0078] According to The constructed signal can be expressed by the following formula:

[0079]

[0080] The Gaussian white noise w l (2f0, t) is set to 20 dB, and similarly, w l (3f0, t) is set to 20 dB. l (13f0, t) is set to 20 dB. l (2f0, t) is set to 20 dB, and similarly, w l (2f0, t) is set to 20 dB, and similarly, w l (3f0, t) is set to 20 dB. l (13f0, t) is set to 20 dB. l (3f0, t) is set to 20 dB. l (13f0, t) is set to 20 dB.

[0081] According to The constructed signal can be expressed by the following formula:

[0082]

[0083] According to The constructed signal can be expressed by the following formula:

[0084]

[0085] According to The constructed signal can be expressed by the following formula:

[0086]

[0087] S4, assuming a virtual array with an element spacing d: The virtual array The corresponding virtual array signal X(t) is composed of x(2f0, t), x(3f0, t), x(13f0, t) and part of the hole elements, where the hole elements are elements with a value of 0, and the composition relationship is:

[0088]

[0089] where X k (t) refers to the virtual array signal X(t) corresponding to the virtual array The signal received by the element at position kd in the virtual array k (fz , t) is the signal vector x(f z , t) corresponds to the virtual array The signal received by the element at position kd in the array, for the signal x k (t) has three cases:

[0090] If the element at position kd corresponds to a single signal x(f z , t), i.e. (f z / f0)(n+2m) = k has only one integer solution, then x k (t) = x k (f z , t). In this embodiment, when k = {2, 3, 4, 8, 9, 12, 13, 26, 39, 52}, the signal is constructed using x k (t) = x k (f z , t).

[0091] If the element at position kd corresponds to multiple signals x(f z , t), i.e. (f z / f0)(n+2m) = k has two integer solutions, then x k (t) = x k (2f0, t). In this embodiment, when k = {6}, the signal is constructed using x k (t) = x k (2f0, t).

[0092] If the element at position kd has no corresponding signal x(f z , t), i.e. (f z / f0)(n+2m) = k has no integer solution, then x k (t) is set to zero. In this embodiment, when k ∈ {1, 2, 3,..., 52} and , the signal is constructed using x k (t) = 0.

[0093] S5, the signal construction is completed, the virtual array signal x(t) is sampled for T fast snaps, where T = 100, and the second moment Suppose a complex matrix represents a matrix with v as the first row, v is a vector to be solved with a dimension of 52 × 1, and the second moment R XX is defined as an observation matrix, and the completed matrix is defined as the original matrix An RXX The observation mask matrix B of the same dimension: R XX The position corresponding to the missing element in R is defined as the hole element, represented by 0. XX An existing element is defined as the position corresponding to the observed element being represented by 1. The observation mask matrix B in this embodiment is shown in the appendix. Figure 5 , attached Figure 5 The black squares represent the observation elements, and the white squares represent the hole elements.

[0094] S6. Achieve the desired result from the observation matrix R by solving the following convex optimization problem. XX Obtain the completed matrix

[0095] Among them, ||.|| F It refers to the Frobenious norm. Represents finding the matrix The trace, ζ is the regularization coefficient, with a value of 0.5. Representative matrix It is positive semi-definite. "⊙" represents the Hadamard product. The convex optimization problem is solved using the CVX toolbox.

[0096] S7, to DOA estimation is performed. In this embodiment, the MUSIC algorithm is used for DOA estimation. The specific steps are as follows: Calculate... The eigenvalues ​​and eigenvectors are sorted, the eigenvalues ​​are arranged in ascending order, and the corresponding eigenvectors are also sorted. Columns K+1 to 51 of the sorted matrix are extracted to form the noise space En.

[0097] Construct a spatial spectrum search function: P(θ) = 1 / |a(θ)′*En*En′*a(θ)|, controlling θ to perform spectral peak search in increments of Δθ = 0.02 degrees from -90 degrees to 90 degrees, where a(θ) is a virtual array. array manifold, p a ∈{1, 2, 3, ..., 52}, with log 10 The spatial spectrum is plotted with P(θ) as the ordinate and θ as the abscissa. The spatial spectrum of the simulation results is shown in the appendix. Figure 6 The dashed line represents θ. k The spectral peaks are the estimated angles. As can be seen, the spectral peak is located near the dashed line with no significant deviation. The experimental results show that 26 incoherent target orientation angles were resolved, proving that the virtual array constructed by the multi-frequency nested MIMO array can effectively resolve target orientation angles far exceeding its own array element number.

[0098] Example 2

[0099] Supplemental to the above embodiments: this embodiment demonstrates the performance of the multi-frequency nested MIMO array under different snapshot numbers and signal-to-noise ratios. It is assumed that there are 9 uncorrelated far-field targets, and the target direction angle is θ k ∈{-20, -15, -10,..., 15, 20}, and the other multi-frequency nested MIMO array position parameters and transmission frequency parameters and the constructed virtual array parameters are the same as those in Embodiment 1. For performance analysis, this embodiment uses the root mean square error (RMSE) of G times Monte Carlo experiments to measure the performance error, where G = 100, and the Monte Carlo experiment is as follows:

[0100] First, replace the signal-to-noise ratio snr in Embodiment 1 with -20 and the snapshot number T with 50, repeat the steps of Embodiment 1 to obtain the spatial spectrum, perform spectrum peak search, and obtain the abscissa of each spectrum peak where g represents the number of Monte Carlo experiments, and 100 Monte Carlo experiments are repeated. In each experiment, the reflection coefficient s k (t) needs to be randomly generated again. The randomly generated s k (t) also obeys a Gaussian random distribution with a mean of 0 and a variance of 1. The RMSE is calculated according to the following formula:

[0101]

[0102] As above, when the snapshot number is T ∈ {50, 80, 120} and the signal-to-noise ratio is snr = {-20, -15,..., 15, 20}. All combinations of snapshot number T and signal-to-noise ratio snr exist, a total of 27 combinations, 100 Monte Carlo experiments are performed for each combination, and the RMSE is calculated. All results are plotted in the attached Figure 7 , the attached Figure 7 with the signal-to-noise ratio as the abscissa, the log 10 RMSE as the ordinate, and different snapshot numbers as data point symbols, a simulation performance graph is drawn. The simulation performance graph shows that the present application performs better under high signal-to-noise ratio conditions, and the more the snapshot number, the better the performance.

[0103] The above embodiments are preferred embodiments of the present application, but the embodiments of the present application are not limited by the above embodiments, and any changes, modifications, substitutions, combinations, simplifications made without departing from the spirit and principles of the present application are equivalent replacement methods and are included in the protection scope of the present application.

Claims

1. A DOA estimation method based on a multi-frequency nested MIMO array, characterized in that, The DOA estimation method includes the following steps: S1. Based on the wave velocity c and frequency f0 of the propagation medium, the wavelength λ = c / f0 is obtained. The minimum unit of the spacing between the elements of the nested MIMO array is set to half a wavelength, i.e. d = 0.5λ = 0.5c / f0. S2. Determine the number of receiver array elements M and the number of transmitter array elements N in the nested MIMO array; where M is the number of receiver array elements and N is the number of transmitter array elements, and the set of transmitter array positions is: Set of receiver array locations: S3, The transmit array of the nested MIMO array has a transmit frequency of f. z A multi-frequency orthogonal probe wave ∈{2f0,3f0,(3MN+1)f0} is used. A nested MIMO array receiver array receives and matches the probe wave echo reflected from the target, generating multiple sets of received signal vectors x(f z ,t): x(f z ,t)={x(2f0,t),x(3f0,t),x((3MN+1)f0,t)} Where x(2f0,t), x(3f0,t), and x((3MN+1)f0,t) are the first, second, and third received signal components, respectively; S4, receive x(f) z The virtual array signal X(t) is reconstructed from the array element spacing d. In step S4, it is assumed that the virtual array has an element spacing of d. Virtual Array The corresponding signal X(t) is given by x(f) z It consists of ,t) and some hole elements, where the hole elements are elements with a value of 0. The composition relationship is as follows: Where X k (t) refers to the virtual array corresponding to the signal vector X(t). The signal received by the array element at position kd, x k (f z ,t) refers to the signal vector x(f) z The virtual array corresponding to t) The signal received by the array element at position kd, for signal X k (t) There are three possible scenarios: If the array element at position kd corresponds to x(f) z A single signal of (f,t), i.e., (f z / f0)(n+Nm)=k has only one integer solution, X k (t)=x k (f z ,t); If the array element at position kd corresponds to x(f) z For multiple signals of type t, for signals with position kd simultaneously existing in signal vectors x(2f0,t), x(3f0,t) and x((3MN+1)f0,t), i.e. (f z / f0)(n+Nm)=k has two integer solutions, X k (t)=x k (2f0,t); If the array element at position kd is in x(f z There is no corresponding signal for (f,t), i.e., (f z / f0)(n+Nm)=k has no integer solutions, that is It means and when Then directly X k (t) is set to zero; For the set of values ​​of the cooperative array; S5. Calculate the second moment R of the virtual array signal X(t). XX ; S6, regarding the second moment R XX Perform low-rank matrix completion to obtain S7, to Perform DOA estimation.

2. The DOA estimation method based on a multi-frequency nested MIMO array according to claim 1, characterized in that, In step S3, the set of positions of the summation cooperative array corresponding to the first received signal component x(2f0,t) is as follows: The set of positions of the summation cooperative array corresponding to the second received signal component x(3f0,t) is as follows: The set of positions of the summation cooperative array corresponding to the third received signal component x((3MN+1)f0,t) is as follows: The set of values ​​for the summation cooperative array is 3. The DOA estimation method based on a multi-frequency nested MIMO array according to claim 1, characterized in that, In step S5, the second moment R XX It is presented as a sparse matrix.

4. The DOA estimation method based on a multi-frequency nested MIMO array according to claim 1, characterized in that, In step S6, a complex matrix with Toplitz and Hermitian matrix characteristics is assumed. This represents a matrix with v as the first row, and the second-order matrix R... XX Defined as the observation matrix, the completed matrix is ​​defined as the original matrix. Construct a with R XX The observation mask matrix B of the same dimension: R XX The positions corresponding to missing elements in RXX are represented by 0, and the positions corresponding to existing elements in RXX are represented by 1. Then, the following convex optimization problem is solved to obtain the position of the observation matrix RXX. xx Obtain the completed matrix ‖.‖ F It refers to the Frobenious norm. Represents finding the matrix The trace, where ζ is the regularization coefficient, ranging from 0 to 1. Representative matrix It is a semi-positive definite form, and ⊙ represents the Hadamard product.

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