Interference detection method for two-dimensional difference frequency DOA estimation

Abstract

The invention discloses an interference detection method for two-dimensional difference frequency DOA (direction of arrival) estimation, which comprises the following steps of: obtaining a noiseless version of an augmented covariance matrix based on a virtual array by utilizing the structural characteristics of a parallel extended co-prime array through structuring a Toeplitz matrix and calling asymptotic maximum likelihood estimation of a structured covariance matrix. Time samples are integrated into a sample covariance matrix, a MUSIC spectrum of each difference frequency is obtained, the direction of a signal source is searched in a noise subspace, and a unique azimuth angle and a unique pitch angle are correspondingly matched. According to the method, a sparse array structure of a double-parallel extended co-prime array is adopted, a difference frequency Hadamard product principle is utilized, a Toeplitz matrix is structured, a difference frequency-based two-dimensional DOA estimation algorithm is realized, the method is applied to multi-target interference detection, a target function is optimized through mathematical modeling so as to obtain a pitch angle and an azimuth angle of an incident signal, and finally algorithm comparison is performed, so that the interference detection accuracy is improved. And quantitatively analyzing the performance difference between the power normalized spectrum of each method and the RMSE and the operation time, and well verifying the feasibility and the excellent performance.

Description

Technical Field

[0001] This invention relates to an interference detection method for two-dimensional difference-frequency DOA estimation, belonging to the field of communication technology. Background Technology

[0002] With the continuous advancement of wireless communication and positioning technologies, Direction of Arrival (DOA) estimation has gradually become an important research direction in the field of signal processing. DOA estimation is a method for determining the direction of a signal source by measuring the phase information of the received signal. This technology is widely used in radar, sonar, communication, and positioning systems, and the accuracy of DOA estimation plays a crucial role in improving system performance.

[0003] Traditional DOA estimation methods include the classic multiple signal classification (MUSIC) algorithm and the Estimation of Signal Parameters via Rotational Invariance Techniques (ESPRIT) algorithm. The MUSIC algorithm is widely used due to its simplicity and high resolution. However, in practical engineering, array antennas are affected by environmental factors, signal interference, and multipath effects during deployment. Improving the accuracy and robustness of DOA estimation, especially in high-dynamic environments and multi-target scenarios, remains a challenge.

[0004] Uniform Linear Arrays (ULAs) are the most common topology in array direction-finding systems. Their elements are deployed at equal intervals, no greater than half the signal wavelength, effectively avoiding phase ambiguity. However, they cannot account for signal sources larger than the number of elements, thus limiting the array's degrees of freedom. In practical applications, the electromagnetic interactions between array elements can cause mutual coupling, and the gridding of sparse signal models can lead to off-grid errors, resulting in decreased accuracy or even failure of DOA estimation. Therefore, researchers have proposed the concept of sparse arrays, which can overcome the element spacing limitations and achieve underdetermined DOA estimation of signals. To address non-ideal conditions such as mutual coupling between elements, off-grid errors, and correlation sources, a spatial smoothing algorithm can be used to improve the sparse spectrum fitting algorithm through preprocessing.

[0005] In recent years, two-dimensional DOA estimation algorithms have attracted much attention, particularly for coprime arrays and nested arrays, when the incident target signal source is a coherent source, due to their ability to achieve large degrees of freedom and array aperture in the coarray domain. Coprime arrays are a typical type of sparse array, usually composed of two uniform linear arrays with mutually prime numbers of elements superimposed. Leveraging the complementary positions of the elements, and by rationally selecting different numbers and spacings of element groups, coprime arrays can provide higher directional accuracy and resolution in complex environments. With theoretical advancements, various sparse array structures have been realized, such as biparallel extended coprime arrays and bidirectional extended frequency-division coprime arrays to obtain extended two-dimensional virtual arrays with greater degrees of freedom. Beyond changes in array structure, DOA estimation algorithms are also continuously being optimized. An improved recursive least squares algorithm can be achieved by introducing minimum interference constraints and generalized convex penalty functions into the traditional least squares method. To address the high computational complexity of the MUSIC algorithm in DOA estimation of near-field signals of two-dimensional arrays, a decoherence processing method is implemented using the MUSIC algorithm with dimensionality reduction and the spatial difference Toeplitz algorithm. The signal is reconstructed using the covariance orthogonal matching pursuit (COMP) algorithm based on singular value decomposition (SVD) and the L1-SVD algorithm. This reduces computational complexity and provides better noise resistance while maintaining the accuracy of DOA estimation. Summary of the Invention

[0006] The purpose of this invention is to address the shortcomings and deficiencies of existing technologies by proposing a two-dimensional difference-frequency DOA estimation interference detection method. This method employs a two-dimensional DOA estimation algorithm for the preprocessed covariance matrix of multiple frequencies and their difference frequencies. Unlike classical two-dimensional DOA estimation methods, it utilizes the structural characteristics of a dual-parallel extended coprime array. By structuring the Toeplitz matrix and invoking the asymptotic maximum likelihood estimation of the structured covariance matrix, a noise-free version of the augmented covariance matrix based on a virtual array is obtained. Time samples are integrated into the sample covariance matrix to obtain the MUSIC spectrum for each difference frequency. The direction of the signal source is then located in the noise subspace, corresponding to a unique azimuth and elevation angle.

[0007] The technical solution adopted by this invention to solve its technical problem is: an interference detection method for two-dimensional difference-frequency DOA estimation, which includes the following steps:

[0008] Step 1: Calculate the autocovariance matrix of the uniform linear array 1. And the cross-covariance matrices of uniform linear arrays 1 and 2 For the matrix respectively Perform vectorization and remove duplicate elements;

[0009] Step 2: Find the position of each virtual array element in the covariance matrix Rearrange the corresponding positions in the matrix to construct the corresponding Toeplitz matrix.

[0010] Step 3: Apply the formula The noiseless 2LT matrix R is partitioned into matrices R1 and R2, and the DOA matrix is ​​constructed.

[0011] Step 4: For the matrix Perform eigenvalue decomposition to obtain eigenvalues ​​E H And calculate the estimated The angle between the incident signal direction and the y-axis can be obtained.

[0012] Step 5: Based on the matrix eigenvector BU estimation The angle between the incident signal direction and the x-axis can be obtained. Finally, match the elevation angle of the incident signal. and azimuth

[0013] Beneficial effects:

[0014] 1. This invention employs a sparse array structure of a dual parallel extended coprime array. Utilizing the principle of the Hadamard product of difference frequencies, a structured Toeplitz matrix is ​​constructed to realize a two-dimensional DOA estimation algorithm based on difference frequencies. This algorithm is applied to multi-target interference detection. The objective function is optimized through mathematical modeling to obtain the elevation and azimuth angles of the incident signal. Finally, the algorithms are compared, and the performance differences in power normalized spectrum, RMSE, and running time of each method are quantitatively analyzed.

[0015] 2. This invention has effectively demonstrated its feasibility and excellent performance, and can reduce the complexity of RMSE and time. Attached Figure Description

[0016] Figure 1 This is a schematic diagram of the one-dimensional coprime array structure of the present invention.

[0017] Figure 2 This is a schematic diagram of the double parallel extended coprime array structure of the present invention.

[0018] Figure 3 This is a schematic diagram showing the relationship between RMSE and SNR for each method.

[0019] Figure 4 This diagram illustrates the relationship between RMSE and the number of information sources for each method.

[0020] Figure 5 This is a schematic diagram showing the relationship between RMSE and the number of snapshots for each method.

[0021] Figure 6 The power normalized spectrum analysis diagrams for each method are shown. Detailed Implementation

[0022] The invention will now be described in further detail with reference to the accompanying drawings.

[0023] like Figure 1 As shown, the one-dimensional coprime array of this invention is composed of two nested uniform linear arrays with M and N elements respectively. The two subarrays share the first element, and the element spacing is Md and Nd respectively, where d = λ / 2 represents the unit element spacing and λ is the signal wavelength. M and N are coprime positive integers, and the total number of elements in the coprime array is T = M + N - 1. Taking the first element of the coprime array as the reference point, the position of each element in the coprime array is represented as follows:

[0024] Ρ={Mn|n∈<0,N-1>}∪{Nn|n∈<0,M-1>} (1)

[0025] Extended coprime array is Figure 1 If the number of array elements is expanded to 2M while the rest remain unchanged, then the expanded coprime array has a total of 2M+N-1 array elements. A double-parallel expanded coprime array is formed by placing two identical expanded coprime arrays in parallel. Assume there are K far-field signal sources s(t) = [s1(t),...,s...]. K (t)] T Incident on a double parallel extended coprime array, as shown in the schematic diagram. Figure 2 As shown.

[0026] exist Figure 2 In this system, a Cartesian coordinate system oxyz is established with the first element of the first extended coprime array as the origin o. Two coprime arrays are placed parallel to each other in the plane xoy, with each element arranged parallel to the x-axis. α k and β k Let θ be the angle between the k-th (k = 1, 2, ..., K) incident signal and the x-axis and y-axis, respectively. k , Let be the elevation and azimuth angles of the k-th incident signal. Then the angular relationship is:

[0027]

[0028] The set of element positions O for each uniform linear array is represented as follows:

[0029] O={nMd|n∈<0,N-1>}∪{mNd|m∈<0,2m-1>} (3)

[0030] but Figure 2 The equation for the received signal matrix of the array shown is:

[0031]

[0032] in

[0033]

[0034] x1(t) and x2(t) are the received signal matrices of two uniform arrays, respectively; A1 and A2 respectively contain angle α k The array manifold matrix; n1(t) and n2(t) are Gaussian white noise matrices.

[0035] According to equation (3), the distance difference between any two elements in each uniform linear array can be calculated, and the set of distance differences Δ t It can be represented as

[0036] Δ t ={±nMd-mNd|n∈<0,N-1>,m∈<0,2M-1>}(5)

[0037] This set is a collection The set of continuous distance differences allows us to construct a virtual uniform linear array with 2MN+1 elements. The set of positions P of each virtual element in the uniform linear array can be represented as:

[0038] P={-MNd,…,-d,0,d,…,MNd} (6)

[0039] By varying the scale, the DOA of the interference is defined as (ω,γ)=(πcosα). k ,πcosβ k The array output at time t (t=1,...,L) of L sample snapshots can be expressed as:

[0040]

[0041] in This represents the steering vector of the k-th signal. Let be the complex Gaussian white noise vector at time t. The array output can be simplified as:

[0042] y(t)=As(t)+n(t) (8)

[0043] in, The theoretical covariance matrix obtained from the array output vector is:

[0044] R = E[y(t)y H [t]=ASA H +σ 2 I (10)

[0045] in,

[0046] S=E[s(t)s H [(t)]=diag([s1,s2,…,sK ] T Let represent the source covariance matrix, and diag(·) be the diagonal matrix function. K Let R be the power of the Kth source. The theoretical covariance matrix R is usually approximated by the sample covariance matrix of L sampling snapshots, i.e., R = R / (L * ...2 * (2 * (2 * (2 * (2 * (2 * (2 * * (2 * * (2 *

[0047]

[0048] The vectorized covariance matrix yields the following single-shot measurement vector model for the virtual domain equivalent signal.

[0049]

[0050] in, The augmented array manifold matrix representing the virtual array. For column-wise Kronecker product operation, the power matrix of the incident signal is s = [s1, s2, ..., s]. K ] T vec(·) is the vectorization function. The number of elements in the virtual array constructed using a uniform linear array is 2MN+1. After identifying the repeated elements, based on the position of each virtual array element within them, we can obtain...

[0051]

[0052] According to equation (7), let Δf i =f i ′-f i ,f i ′>f i Using frequency data y fi (t) and its complex conjugate The Hadamard product is obtained. The expression is:

[0053]

[0054] symbol This represents the multiplication operation within the block matrix, compared to formula (7). It is the sum of the guiding vectors, but in Δf i Below, the guiding vector is With frequency {f i ′,f i Different, difference frequency Δf i Lower than the aliasing frequency of the array.

[0055] However, as the processing frequency decreases, Δf i <f iThe resolution of the beamformer will decrease. Conventional beamformers produce a wider main lobe, thus reducing resolution. Furthermore, the cross term in equation (14) will generate unwanted pseudo-DOA in the beamforming spectrum. K real DOA will produce K 2 -K pseudo-DOA values, which are determined by the frequency and the true DOA.

[15] According to formula (11):

[0056]

[0057] Eigenvalue decomposition of the above equation decomposes it into two parts: signal and noise.

[0058]

[0059] Corresponding to the largest K 2 The eigenvector U with eigenvalues signal Zhang Cheng signal subspace. Due to the effect of the cross term (14), the eigenvalues ​​above the noise floor can reach a maximum of K. 2 One. The rest are 2N-K. 2 eigenvectors U noise Zhang establishes a noise subspace, which contains no signal components and is orthogonal to the signal subspace, i.e.:

[0060]

[0061] Define the projection operator onto the noise subspace:

[0062]

[0063] steering vector in any direction Its projected energy in the noise subspace is:

[0064]

[0065] According to formula (19), when ω is the DOAω of the real signal k At that time, there was:

[0066]

[0067] To generate a peak at the true DOA, the difference-frequency MUSIC (DF-MUSIC) method defines the pseudospectrum as the reciprocal of the projected energy of the noise subspace, with the azimuth angle ω. k Then, based on the power spectrum estimation, the expression is:

[0068]

[0069] The DF-MUSIC method requires the use of equally spaced DFs, i.e.:

[0070] i = 1, ..., F, regardless of how the frequency changes with respect to i

[0071] They all share the same steering vector. Because in the true ω k When the denominator is zero (or extremely small), its reciprocal will show a significant peak, thus yielding P. DF-MUSIC (ω k This allows for DOA estimation.

[0072] The position S of the virtual sensor v The expression is:

[0073] S v ={l i -l j |i,j=1,2,…,2N} (22)

[0074] S v The missing element in the middle is called a hole, l = (l x ,l y ) represents the physical sensor specifications. S v The union of the holes is defined as S p .

[0075] Spatial smooth subarrays are used to construct the enhanced covariance matrix. Its expression is:

[0076]

[0077] Among them, the guiding matrix Associated with 2MN+2 virtual sensors. The noise-free enhanced covariance matrix is ​​then used. Divided into four blocks, its expression is:

[0078]

[0079] Where B(ω,γ) represents the guidance matrix consisting of MN+1 sensors, and the expression for the diagonal matrix E is:

[0080] Each block is a Toeplitz matrix, therefore (24) is a second-order Toeplitz matrix, which can be represented as:

[0081]

[0082] The Toeplitz matrix for each block can be constructed using the position vectors of the virtual array elements, where the autocovariance matrix of the uniform linear array 1 is expressed as:

[0083]

[0084] The elements in the formula are conjugate symmetric about the main diagonal, and have the following properties:

[0085] Hermitian property, then equation (26) can be rewritten as

[0086]

[0087] in For including ω k The matrix.

[0088] According to equation (4), the cross-covariance matrix of uniform linear arrays 1 and 2 can be calculated as follows:

[0089]

[0090] right After vectorization, we get:

[0091]

[0092] in In the formula, P2 is the one-dimensional snapshot vector of the cross-covariance matrix after vectorization. Here, a2 is the Kronecker product operator, Q2 is the vectorized array manifold matrix of a uniform linear array of 2, and a2(ω K ,γ K ) is the steering vector of the k-th incident signal, containing azimuth and elevation information. Similarly, Its expression is:

[0093] It also possesses the Hermitian property, so equation (30) can be rewritten as:

[0094]

[0095] in For including ω k and γ k The matrix. At this point, each part of equation (25) has been given.

[0096] Based on the geometric construction relationship in (25), R can be divided into two 2(MN+1)×(MN+1) dimensional signal subspace matrices R1 and R2, forming... Specifically,

[0097] R1 = BSUB H R2 = BSE H UB H (32)

[0098] in B is the guidance matrix without holes. According to R1, we can obtain:

[0099] SUB H = (B H B) -1 B H R1 (33)

[0100] Substituting it into R2, we get:

[0101] R2 = BE H (B H B) -1 B H R1 (34)

[0102] Multiply both sides to the right We can obtain:

[0103]

[0104] From formula (18), we know that vector BU is a matrix. The eigenvectors of , and their corresponding eigenvalues ​​are E H ,Right now:

[0105]

[0106] From formula (19), it can be obtained by adjusting the matrix. We estimate γ by performing eigenvalue decomposition, i.e.:

[0107]

[0108] Define matrix H1 = [I V-1 0 (V-1)×1 ] and matrix H2 = [0 (V-1)×1 I V-1 ], where V is half the order of R, and construct the switching matrix:

[0109]

[0110] Combining with BU, we can conclude that:

[0111] J e BUΩ=J f BU (39)

[0112] From the above formula, we get Ω=(J e BU) + (J f B), used to estimate ω, i.e.:

[0113]

[0114] This invention, based on the principle of two-dimensional DOA estimation for parallel extended coprime arrays, presents the main steps of DOA estimation:

[0115] a) Calculate the autocovariance matrix of the uniform linear array 1. And the cross-covariance matrices of uniform linear arrays 1 and 2 For the matrix respectively Perform vectorization and remove duplicate elements;

[0116] b) Locate the virtual array elements in the covariance matrix. Rearrange the corresponding positions in the matrix to construct the corresponding Toeplitz matrix.

[0117] c) Divide the noise-free 2LT matrix R in formula (25) into matrices R1 and R2 to construct the DOA matrix.

[0118] d) For the matrix Perform eigenvalue decomposition to obtain eigenvalues ​​E H And calculate the estimated The angle between the incident signal direction and the y-axis can be obtained.

[0119] e) According to the matrix eigenvector BU estimation The angle between the incident signal direction and the x-axis can be obtained. Finally, match the elevation angle of the incident signal. and azimuth

[0120] The simulation experiments and performance analysis of this invention include the following:

[0121] To verify the accuracy of the proposed method for DOA estimation, simulations were performed. The simulation used coprime integer arrays, where M = 3 and N = 5, forming a coprime array of 2M + N - 1 physical sensors. The sensor positions were {0, 3, 4, 6, 8, 10, 12, 16, 20, 25}. The root mean square error (RMSE) was used to evaluate the accuracy of the orientation angle estimation as a performance metric, with the Cramer-Rao boundary (CRB) included as a benchmark.

[0122]

[0123] Where M t Indicates the number of Monte Carlo trials. and ω for the mth time k and γ kThe estimated value. The method of this invention selects the least squares method and the MUSIC method.

[17] Enhanced MUSIC Method

[18] In comparison.

[0124] Simulation Experiment 1: Four source points were set up, with angle pairs distributed within the interval [-π, π]. The signal-to-noise ratio (SNR) ranged from -10dB to 10dB, and the number of snapshots was 600. Figure 3 The relationship between RMSE and SNR obtained using different methods is shown.

[0125] Simulation Experiment 2: The number of information sources was set to vary from 2 to 8, the signal-to-noise ratio was 6dB, and the number of snapshots was 800. Figure 4 The relationship between RMSE obtained using different methods and the number of information sources is shown.

[0126] Simulation Experiment 3: Set 4 source points, signal-to-noise ratio of 0dB, and the number of snapshots varies from 200 to 600. Figure 5 The relationship between RMSE obtained using different methods and the number of snapshots is shown.

[0127] Simulation Experiment 4: Two interference sources were set up with azimuth angles of -15° and -20° respectively, frequency range of 50 to 75MHz, sensor spacing of about 3.75m, signal-to-noise ratio of 10dB, number of snapshots of 50, and angular resolution of 0.005. Figure 6 The left side tightly locks two extremely narrow high-energy peaks at -15° and -20°, suppressing background noise to almost below -50dB. The right figure shows the power normalized spectrum obtained using different methods. Two interference signals are incident on the left side, with the upper blue line extending at an angle of -15° and the lower line at -20°. The angle corresponding to the peak in the right figure is the estimated value.

[0128] The basic principle of least squares in DOA estimation is to solve for the angle of arrival (DOA) by minimizing the difference between the observed signal and the model-predicted signal. The MUSIC algorithm determines DOA by finding the mutual orthogonality between the noise subspace and the array response matrix. The method of this invention fully utilizes the structural characteristics of a two-dimensional coprime array and constructs an angle mapping relationship based on the Hadamard product of the second-order Toeplitz matrix and the difference frequency. Therefore, when SNR > -8dB, the DOA estimation accuracy is superior to the other three algorithms. As the number of signal sources increases, the RMSE values ​​estimated by the other three methods increase significantly, while the RMSE value calculated by the method of this invention always approaches the theoretical limit. Even with changes in the signal-to-noise ratio and the number of snapshots, relatively higher DOA estimation accuracy can still be achieved.

[0129] To quantitatively verify the interference detection performance of the algorithm of this invention in DOA estimation, specific numerical analysis was performed on four types of methods.

[0130] Table 1: Power Normalized Spectrum Numerical Analysis Table for Each Method

[0131]

[0132] As shown in the table above, the spectrum obtained by the least squares method has a wide main lobe and high side lobes. The MUSIC algorithm improves the side lobes, and RMUSIC improves the resolution. The algorithm of this invention obtains super-resolution by separating the noise subspace and utilizing the orthogonality of the array manifold. The difference frequency makes the array aperture "expand", further amplifying the advantages. This makes the angle deviation less than 0.2°, the dynamic range improved by about 20dB, the sharper the peak, the higher the resolution, and the lowest the side lobes. It can be used for multi-target interference detection.

[0133] Set SNR = 10dB, and set the other parameters the same as in simulation experiment 1. The performance indicators are shown in the table below.

[0134] Table 2: Comparison of Indicators for Each Method

[0135]

[0136]

[0137] As shown in the table above, the least squares method yields the highest RMSE, but has lower complexity and relatively faster running time. The MUSIC and RMUSCIC algorithms reduce the RMSE, but significantly increase the running time. The algorithm in this invention utilizes a structured Toeplitz matrix, effectively reducing time complexity and achieving difference frequency on a double parallel extended coprime array, ensuring that the RMSE always approaches the CRB.

Claims

1. An interference detection method for two-dimensional difference-frequency DOA estimation, characterized in that, The method includes the following steps: Step 1: Calculate the autocovariance matrix of the uniform linear array 1. And the cross-covariance matrices of uniform linear arrays 1 and 2 For the matrix respectively Perform vectorization and remove duplicate elements; Step 2: Find the position of each virtual array element in the covariance matrix Rearrange the corresponding positions in the matrix to construct the corresponding Toeplitz matrix. Step 3: Apply the formula The noiseless 2LT matrix R is partitioned into matrices R1 and R2, and the DOA matrix is ​​constructed. Step 4: For the matrix Perform eigenvalue decomposition to obtain eigenvalues ​​E H And calculate the estimated The angle between the incident signal direction and the y-axis can be obtained. Step 5: Based on the matrix eigenvector BU estimation The angle between the incident signal direction and the x-axis can be obtained. Finally, match the elevation angle of the incident signal. and azimuth 2. The interference detection method for two-dimensional difference-frequency DOA estimation according to claim 1, characterized in that, Step 1 includes: a one-dimensional coprime matrix is ​​composed of two nested uniform linear arrays with M and N elements respectively. The two subarrays share the first element, and the element spacings are Md and Nd respectively, where d = λ / 2 represents the unit element spacing, λ is the signal wavelength, M and N are coprime positive integers, and the total number of elements in the coprime matrix is ​​T = M + N - 1. Taking the first element of the coprime matrix as a reference point, the position of each element in the coprime matrix is ​​represented as follows: Ρ={Mn|n∈<0,N-1>}∪{Nn|n∈<0,M-1>} (1) An extended coprime array is formed by expanding the number of array elements to 2M while keeping the rest unchanged. Therefore, an extended coprime array has a total of 2M+N-1 array elements. A double-parallel extended coprime array is formed by placing two identical extended coprime arrays in parallel. Let there be K far-field signal sources s(t) = [s1(t),...,s...]. K (t)] T Incident onto a double parallel extended coprime array.

3. The interference detection method for two-dimensional difference-frequency DOA estimation according to claim 1, characterized in that, Step 2 includes the following: the set of positions P of each virtual element in the uniform linear array can be represented as: P={-MNd,…,-d,0,d,…,MNd} (6) By varying the scale, the DOA of the interference is defined as (ω,γ)=(πcosα). k ,πcosβ k The array output at time t (t=1,...,L) of L sample snapshots can be expressed as: in This represents the steering vector of the k-th signal. Let be the complex Gaussian white noise vector at time t. The array output can be simplified as: y(t)=As(t)+n(t) (8) in, The theoretical covariance matrix obtained from the array output vector is: R=E[y(t)y H (t)]=ASA H +σ 2 I (10) in, S=E[s(t)s H [(t)]=diag([s1,s2,…,s K ] T ) represents the source covariance matrix, diag(·) is the diagonal matrix function, and s K Let R be the power of the Kth source. The theoretical covariance matrix R is usually approximated by the sample covariance matrix of L sampling snapshots, i.e., R = R / (L * L ... The vectorized covariance matrix yields the following single-shot measurement vector model for the virtual domain equivalent signal. in, A represents the enhanced array manifold matrix of the virtual array. For column-wise Kronecker product operation, the power matrix of the incident signal is s = [s1, s2, ..., s]. K ] T vec(·) is the vectorization function. The number of elements in the virtual array constructed using a uniform linear array is 2MN+1. After identifying the repeated elements, based on the position of each virtual array element within them, we can obtain... According to equation (7), let Δf i =f i ′-f i ,f i ′>f i Using frequency data Its conjugate The Hadamard product is obtained. The expression is: symbol This represents the multiplication operation within the block matrix, compared to formula (7). It is the sum of the guiding vectors, but in Δf i Below, the guiding vector is With frequency {f i ′,f i Different, difference frequency Δf i Lower than the aliasing frequency of the array.

4. The interference detection method for two-dimensional difference-frequency DOA estimation according to claim 1, characterized in that: Step 4 Eigenvalue decomposition is performed to separate it into signal and noise components: Corresponding to the largest K 2 The eigenvector U with eigenvalues signal In the Zhang Cheng signal subspace, due to the effect of the cross term (14), the eigenvalues ​​above the noise floor can reach a maximum of K. 2 One, the rest are 2N-K 2 eigenvectors U noise Zhang establishes a noise subspace, which contains no signal components and is orthogonal to the signal subspace, i.e.: Define the projection operator onto the noise subspace: steering vector in any direction Its projected energy in the noise subspace is According to formula (19), when ω is the DOAω of the real signal k At that time, there was:

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