A fourth-order cumulant DFT taylor compensation based direct positioning method

Abstract

The application discloses a kind of based on fourth-order cumulant DFT taylor compensation direct positioning method, it is related to array signal processing technical field, through observation station receives non-gaussian signal from K target radiation source, and each observation station is equipped with sparse nested array, according to received signal, the fourth-order cumulant of it is calculated, the spatial smoothing and vectorization processing of fourth-order cumulant matrix obtains the received signal of continuous virtual array element, constructs DFT algorithm spectrum function and obtains the initial position coordinate estimation of signal source, then using the array aperture of virtual array after vectorization, through taylor compensation, the taylor compensation result of the position of K radiation source is obtained.The application uses fourth-order cumulant, compared with traditional second-order cumulant, more information is included, and has blind Gaussian characteristic, array expansion characteristic, phase detectability;Using taylor compensation improves resolution and estimation accuracy.

Description

Technical Field

[0001] This invention relates to the field of array signal processing technology, and in particular to a direct localization method based on fourth-order cumulant DFT Taylor compensation. Background Technology

[0002] In recent years, a new type of sparse array has attracted widespread attention and research. Sparse nested arrays are a typical example of sparse arrays. Compared with traditional uniform linear arrays, they can achieve larger array apertures and higher degrees of freedom, thus having advantages in spectral estimation accuracy and resolution. Furthermore, they can effectively reduce the mutual coupling effect between array elements and have lower hardware costs compared with general uniform arrays.

[0003] In practice, the signal incident on the array is a non-Gaussian signal, characterized by its non-Gaussian distribution. In such cases, the traditional second-order cumulants are insufficient to describe all the information in the signal. Therefore, fourth-order cumulants are used to obtain all the signal's characteristics. Compared to second-order cumulants, fourth-order cumulants not only have blind properties against Gaussian noise but also expand the array elements. When using them for parameter estimation, a more accurate received signal matrix can be obtained, leading to more precise estimation results.

[0004] This method utilizes the direct localization algorithm to perform multi-array information fusion directly at the raw received data layer, eliminating the need for intermediate parameter estimation and directly extracting the radiation source's location estimation information from the fused data. This approach fully leverages the correlation between data received from different locations, avoiding secondary error propagation caused by intermediate parameter estimation, thereby significantly improving localization performance. Summary of the Invention

[0005] To address the above technical problems, this invention provides a direct localization method based on fourth-order cumulant DFT Taylor compensation, comprising the following steps:

[0006] S1. Construct a sparse nested array in the two-dimensional plane xy, and receive non-Gaussian signals from K target radiation sources through the sparse nested arrays equipped at each observation station;

[0007] S2. Calculate the fourth-order cumulant of the non-Gaussian signal received by the sparse nested array to obtain the R4 matrix. Vectorize the R4 matrix and sort and remove redundancy to obtain the Z matrix.

[0008] S3. Construct the DFT algorithm spectrum function to obtain the initial position coordinates of each target radiation source;

[0009] S4. Using the array aperture of the virtual array, Taylor compensation is performed to obtain the Taylor compensation results for the positions of the K radiation sources.

[0010] The technical solution further defined in this invention is:

[0011] Furthermore, in step S1, the sparse nested array consists of two uniform subarrays with N1 and N2 elements respectively, and the total number of elements M = N1 + N2. The element spacing of the first subarray is d, and the element spacing of the second subarray is (N1 + 1)d, where d = λ / 2 is half the wavelength. The positions of all physical elements are represented as follows:

[0012]

[0013] The coordinates of the K radiation sources are p k =(x k y k ) T The location coordinates of the observation stations k = 1, 2, ..., K, L are u l =(x l y l ) T If l = 1, 2, ..., L, then the azimuth angle of the k-th radiation source incident on the l-th observation station is θ. k，l , represented as

[0014]

[0015] At time t, the signals received by the sparse nested array of the l-th observation station from the K target radiation sources are represented as:

[0016] x l (t)=A l s l (t)+n l (t)

[0017] in, Let represent the direction matrix, which is an M×K dimensional matrix composed of the direction vectors of K radiation sources; where the direction vector of the k-th radiation source is . λ represents the wavelength; d1, ..., d M These represent the positions of each physical array element; Let J represent the zero-mean non-Gaussian signal vector from the k-th radiation source to the 1st observation station, and satisfy 1≤t≤J, where J is the total number of snapshots; This indicates that the mean is zero and the variance is... The additive noise vector, where the noise and signal are independent, is Gaussian white noise, (·) T This is a transpose.

[0018] In the aforementioned direct localization method based on fourth-order cumulant DFT Taylor compensation, step S2 uses fourth-order cumulants to characterize signal features. The definition of the fourth-order cumulant for a known zero-mean N-dimensional stationary random process x is as follows:

[0019]

[0020] Where cum(·) represents the operation of calculating the cumulative amount, (·) * Indicates conjugate, therefore the received signal x l The fourth-order cumulant of (t) is calculated by the following formula:

[0021]

[0022] in, It can be seen that a 4，x (θ k ) by a(θ k )constitute, s k The fourth-order cumulant of (t) is expressed as To obtain the received signal from the virtual array, R4 is virtualized to obtain the virtual received signal from the l-th observation station:

[0023] z l =vec(R4)

[0024] in, Let T be a single snapshot signal obtained from a virtual array, and let T be z. l The length.

[0025] The aforementioned direct localization method based on fourth-order cumulant DFT Taylor compensation includes the following sub-steps in step S3.

[0026] S3.1 Construct a search grid near the radiation source, searching in both the x and y directions. x , l y Next, if the coordinates of the search grid points are (x, y), then for the grid points near the k-th radiation source, calculate the azimuth angle of each grid point relative to the 1st observation station:

[0027]

[0028] S3.2, the direction vector of the kth signal in the virtual array is a v (θ l，k ), where k = 1, 2, ..., K, and the direction vector after DFT processing of the signal is: Its q-th element is

[0029]

[0030] From the above formula, we can see that when q k =T sinθ l，k When / 2 is an integer There is one and only the qth kWhen q is not zero, k When not an integer, Only q k The elements in the neighborhood are not zero, and all other elements are zero, therefore we search for... The coarse estimate of the position q of the non-zero element k , for θ l，k Perform initial estimation;

[0031] S3.3, By analyzing the virtual received signal z l A DFT transform is performed to obtain the angle estimate. The positions of the K largest peaks of the received signal vector after the DFT transform are denoted as... Then the initial angle estimation Represented as Therefore, we get:

[0032]

[0033] sint l Substituting into the above formula, we get:

[0034]

[0035] S3.4 Constructing the normalized DFT matrix Its t-th element is represented as:

[0036]

[0037] Let f k,l =f k，l z l f obtained from L observation stations k,l =F k，l z l Adding l = 1, ..., L together, we get:

[0038]

[0039] Will Substituting into the above formula, we get:

[0040]

[0041] Where z(t) represents the t-th element of the single snapshot vector z, t = 1, 2, ..., T; find f k The maximum value is the rough estimate of the location of the k-th radiation source.

[0042] The aforementioned direct localization method based on fourth-order cumulant DFT Taylor compensation includes the following sub-steps in step S4.

[0043] S4.1 The coarse estimation results of the positions of the K radiation sources obtained by searching the spectral function constructed by the DFT algorithm are as follows: Define the coarse estimation vector on the x-axis Coarse estimate vector on the y-axis After calculating the fourth-order cumulant of the non-Gaussian radiation source, a virtual uniform linear array is obtained, and the positions of its virtual elements are represented by a set. To indicate:

[0044]

[0045] Where T is the total number of elements in the virtual array, i.e., the length of vector z, and

[0046] S4.2, Let the equivalent steering matrix of the first virtualized observation station be... and The equivalent steering vector is:

[0047] in,

[0048]

[0049] And d v v = 1, 2, ..., T represents The v-th element, The x-coordinate represents the coarse estimate of the k-th radiation source. The y-coordinate represents the coarse estimate of the k-th radiation source;

[0050] S4.3, to exist Performing a first-order Taylor series expansion at the given point and neglecting second-order and higher error terms, we obtain:

[0051]

[0052] Where, p k，l (1) and p represents the x-coordinate of the radiation source and the x-coordinate of the coarse estimation result, respectively. k，l (2) and These represent the y-coordinates of the radiation source and the coarse estimation result, respectively.

[0053] S4.4, Definition of Λ x =diag(ξ x ), Λ y =diag(ξ y ),and The virtual received signal of the first observation station is rewritten as:

[0054]

[0055] but

[0056]

[0057] The above expression can be written as:

[0058]

[0059] in,

[0060] S4.5, According to S=B + Given Z, calculate S and divide it into blocks S1, S2, and S3. S1 represents rows 1 to K of matrix S, S2 represents rows K+1 to 2K of matrix S, and S3 represents rows 2K+1 to 3K of matrix S.

[0061] S4.6 The Taylor compensation values ​​for the x-axis and y-axis of the radiation source location estimation results are as follows:

[0062] Λ x =diag(ξ x ) = S2. / S1

[0063] Λ y =diag(ξ y ) = S3. / S1

[0064] The final location results of the K radiation sources are:

[0065]

[0066]

[0067] in, This represents the coarse estimate vector on the x-axis. This represents a coarse estimate vector on the y-axis.

[0068] The aforementioned direct localization method based on fourth-order cumulant DFT Taylor compensation also includes a method for proving the effectiveness of steps S1 to S4, specifically: verification is performed through MATLAB simulation analysis, using root mean square error (RMSE) as the performance evaluation criterion. The RMSE is defined as follows:

[0069]

[0070] Where K is the number of target radiation sources, MN is the number of Monte Carlo simulation experiments, and mn represents the mn-th Monte Carlo simulation experiment. and Let x and y be the coarse estimates of the x and y coordinates of the k-th radiation source obtained from the mn-th test, respectively. k and y k These are the x and y coordinates of the radiation source, respectively.

[0071] The beneficial effects of this invention are:

[0072] (1) In this invention, the advantages of sparse nested arrays, such as larger aperture and higher spatial degree of freedom, are utilized to effectively improve the positioning accuracy of the direct positioning algorithm.

[0073] (2) In this invention, the fourth-order cumulant is used to process the received data of the antenna array, which can more completely characterize all the statistical characteristics of the target signal. Furthermore, the fourth-order cumulant is not sensitive to Gaussian noise and can suppress Gaussian white noise to the greatest extent compared with the second-order statistic, while effectively expanding the array aperture.

[0074] (3) In this invention, the DFT algorithm can be used directly for virtual signals of a single snapshot without the need for a decorrelation process, thus avoiding spatial smoothing and reducing complexity; at the same time, Taylor compensation is used to improve resolution and estimation accuracy. Attached Figure Description

[0075] Figure 1 This is a schematic diagram of the overall process of the present invention;

[0076] Figure 2 This is a scatter plot of the DFT-DPD algorithm used in this embodiment of the invention.

[0077] Figure 3 This is a schematic diagram illustrating the performance changes of the method in this embodiment of the invention under different signal-to-noise ratios;

[0078] Figure 4 This is a schematic diagram illustrating the performance changes of the method of this invention under different snapshot numbers. Detailed Implementation

[0079] This embodiment provides a direct localization method based on fourth-order cumulant DFT Taylor compensation, such as... Figure 1 As shown, it includes the following steps

[0080] S1. Construct a sparse nested array in the two-dimensional plane xy, and receive non-Gaussian signals from K target radiation sources through the sparse nested arrays provided by each observation station.

[0081] A sparse nested array consists of two uniform subarrays with N1 and N2 elements respectively, and the total number of elements is M = N1 + N2. The element spacing of the first subarray is d, and the element spacing of the second subarray is (N1 + 1)d, where d = λ / 2 is half the wavelength. The positions of all physical elements are represented as follows:

[0082]

[0083] The coordinates of the K radiation sources are p k =(x k y k ) T The location coordinates of the observation stations k = 1, 2, ..., K, L are u l =(x l y l ) T If l = 1, 2, ..., L, then the azimuth angle of the k-th radiation source incident on the l-th observation station is θ. k，l , represented as

[0084]

[0085] At time t, the signals received by the sparse nested array of the l-th observation station from the K target radiation sources are represented as:

[0086] x l (t)=A l s l (t)+n l (t)

[0087] in, Let represent the direction matrix, which is an M×K dimensional matrix composed of the direction vectors of K radiation sources; where the direction vector of the k-th radiation source is . λ represents the wavelength; d1, ..., d M These represent the positions of each physical array element; Let J represent the zero-mean non-Gaussian signal vector from the k-th radiation source to the 1st observation station, and satisfy 1≤t≤J, where J is the total number of snapshots; This indicates that the mean is zero and the variance is... The additive noise vector, where the noise and signal are independent, is Gaussian white noise, (·) T This is a transpose.

[0088] S2. Calculate the fourth-order cumulant of the non-Gaussian signal received by the sparse nested array to obtain the R4 matrix. Vectorize the R4 matrix and sort and remove redundancy to obtain the Z matrix.

[0089] Since the array receives a non-Gaussian signal, the autocovariance matrix in this case can no longer fully express all the statistical characteristics of the target radiation source. Therefore, a higher-order fourth-order cumulant is needed to characterize the signal features. The definition of the fourth-order cumulant for a zero-mean N-dimensional stationary random process x is as follows:

[0090]

[0091] Where cum(·) represents the operation of calculating the cumulative amount, (·) * Indicates conjugate, therefore the received signal x l The fourth-order cumulant of (t) is calculated by the following formula:

[0092]

[0093] in, It can be seen that a 4,x (θ k ) by a(θ k )constitute, s k The fourth-order cumulant of (t) is expressed as To obtain the received signal from the virtual array, R4 is virtualized to obtain the virtual received signal from the l-th observation station:

[0094] z l =vec(R4)

[0095] in, Let T be a single snapshot signal obtained from a virtual array, and let T be z. l The length.

[0096] S3. Construct the DFT algorithm spectral function to obtain the initial position coordinates of each target radiation source, which includes the following steps:

[0097] S4.1 Construct a search grid near the radiation source, searching in both the x and y directions. x , l y Next, if the coordinates of the search grid points are (x, y), then for the grid points near the k-th radiation source, calculate the azimuth angle of each grid point relative to the l-th observation station:

[0098]

[0099] Among them, sin t l This represents the azimuth angle of each grid point relative to the first observation station.

[0100] S3.2, the direction vector of the kth signal in the virtual array is a v (θ l，k ), where k = 1, 2, ..., K, and the direction vector after DFT processing of the signal is: Its q-th element is

[0101]

[0102] From the above formula, we can see that when q k =T sinθ l，k When / 2 is an integer There is one and only the qth k When q is not zero, k When not an integer, Only q k The elements in the neighborhood are not zero, and all other elements are zero, therefore we search for... The coarse estimate of the position q of the non-zero element k , for θ l，k Make an initial estimate.

[0103] S3.3 In direct positioning applications, the azimuth angle and its corresponding azimuth vector are problems that need to be solved. This is achieved by analyzing the virtual received signal z... l A DFT transform is performed to obtain the angle estimate. The positions of the K largest peaks of the received signal vector after the DFT transform are denoted as... Then the initial angle estimation Represented as Therefore, we get:

[0104]

[0105] sint l Substituting into the above formula, we get:

[0106]

[0107] in, This represents the positions of the K largest peaks in the received signal vector after the DFT transformation.

[0108] S3.4 Constructing the normalized DFT matrix Its t-th element is represented as:

[0109]

[0110] Let f k,l =f k，l z l f obtained from L observation stations k,l =F k，l z l Adding l = 1, ..., L together, we get:

[0111]

[0112] Will Substituting into the above formula, we get:

[0113]

[0114] Where z(t) represents the t-th element of the single snapshot vector z, t = 1, 2, ..., T; find f kThe maximum value is the rough estimate of the location of the k-th radiation source.

[0115] S4. Using the array aperture of the virtual array, Taylor compensation is performed to obtain the Taylor compensation results for the positions of the K radiation sources. This includes the following steps:

[0116] S4.1 The coarse estimation results of the positions of the K radiation sources obtained by searching the spectral function constructed by the DFT algorithm are as follows: Define the coarse estimation vector on the x-axis Coarse estimate vector on the y-axis After calculating the fourth-order cumulant of the non-Gaussian radiation source, a virtual uniform linear array is obtained, and the positions of its virtual elements are represented by a set. To indicate:

[0117]

[0118] Where T is the total number of elements in the virtual array, i.e., the length of vector z, and

[0119] S4.2, Let the equivalent steering matrix of the virtualized l-th observation station be . and The equivalent steering vector is:

[0120] in,

[0121]

[0122] And d v v = 1, 2, ..., T represents The v-th element, The x-coordinate represents the coarse estimate of the k-th radiation source. The y-coordinate represents the coarse estimate of the k-th radiation source.

[0123] S4.3, to exist Performing a first-order Taylor series expansion at the given point and neglecting second-order and higher error terms, we obtain:

[0124]

[0125] Where, p k，l (1) and p represents the x-coordinate of the radiation source and the x-coordinate of the coarse estimation result, respectively. k，l (2) and These represent the y-coordinates of the radiation source and the coarse estimation result, respectively.

[0126] S4.4, Definition of Λ x=diag(ξ x ), Λ y =diag(ξ y ),and The virtual received signal of the l-th observation station is rewritten as:

[0127]

[0128] but

[0129]

[0130] The above expression can be written as:

[0131]

[0132] in,

[0133] S4.5, According to S=B + Given Z, calculate S and divide it into blocks S1, S2, and S3. S1 represents rows 1 to K of matrix S, S2 represents rows K+1 to 2K of matrix S, and S3 represents rows 2K+1 to 3K of matrix S.

[0134] S4.6 The Taylor compensation values ​​for the x-axis and y-axis of the radiation source location estimation results are as follows:

[0135] Λ x =diag(ξ x ) = S2. / S1

[0136] Λ y =diag(ξ y ) = S3. / S1

[0137] The final location results of the K radiation sources are:

[0138]

[0139]

[0140] in, This represents the coarse estimate vector on the x-axis. This represents a coarse estimate vector on the y-axis.

[0141] To demonstrate the effectiveness of the algorithm in this embodiment, MATLAB simulation analysis is used. Root mean squared error (RMSE) is used as the performance evaluation criterion, defined as follows:

[0142]

[0143] Where K is the number of target radiation sources, MN is the number of Monte Carlo simulation experiments, and mn represents the mn-th Monte Carlo simulation experiment. and Let x and y be the coarse estimates of the x and y coordinates of the k-th radiation source obtained from the mn-th test, respectively. k and y k These are the x and y coordinates of the radiation source, respectively.

[0144] like Figure 2 The image shows a scatter plot of the direct localization method based on fourth-order cumulant DFT Taylor compensation proposed in this embodiment. In the simulation, it is assumed that the number of elements in the two subarrays of the sparse nested array is N1 = N2 = 3, then the total number of elements is M = 6. The element spacing of the first subarray is d, and the element spacing of the second subarray is (N1+1)d, where d = λ / 2 is half the wavelength. The number of target radiation sources is K = 2, with position coordinates of (-500.5, 600.5) and (800.5, 200.5), and the signal-to-noise ratio is set to 5dB, with the number of snapshots set to J = 200. The scatter plot of the simulation results shows that the position coordinates of the target radiation sources obtained by the DFT-DPD method described in this embodiment are near the preset target source position coordinates, demonstrating the effectiveness of the algorithm used in this embodiment and its ability to directly locate the target radiation sources.

[0145] like Figure 3 The figure shows the performance variation of the direct localization method based on fourth-order cumulant DFT Taylor compensation proposed in this embodiment under different signal-to-noise ratios. In the simulation, it is assumed that the number of elements in the two subarrays of the sparse nested array is N1 = N2 = 3, then the total number of elements is M = 6. The element spacing of the first subarray is d, and the element spacing of the second subarray is (N1+1)d, where d = λ / 2 is half the wavelength. The number of target radiation sources K = 2, with position coordinates of (-500.5, 600.5) and (800.5, 200.5). The signal-to-noise ratio (SNR) of each observation station node starts from 0dB and steps in 5dB increments to 25dB. In this invention, the number of Monte Carlo simulation experiments MN = 500, and the number of snapshots J = 200.

[0146] The simulation results show that: (1) Since the SDF-DPD and Capon-DPD algorithms use the covariance matrix of the received signal, which is a second-order cumulant, while the DFT-DPD algorithm uses the fourth-order cumulant of the received signal and takes advantage of the non-Gaussian nature of the radiated signal, under this condition, the positioning performance of DFT-DPD is better than that of SDF-DPD and Capon-DPD algorithms; (2) Since the radiation source is not on the search grid point, the positioning performance of SDF-DPD, Capon-DPD and DFT-DPD algorithms will have a bottleneck effect when the SNR is large, while the performance of DFT-DPD algorithm after Taylor compensation is significantly improved; (3) When the number of snapshots is constant, the higher the signal-to-noise ratio, the better the estimation performance of the method proposed in this invention. Therefore, the positioning accuracy of the DFT-DPD method described in this embodiment can be improved by appropriately increasing the signal-to-noise ratio.

[0147] like Figure 4 The figure shows the performance variation of the direct localization method based on fourth-order cumulant DFT Taylor compensation proposed in this embodiment under different snapshot numbers. Figure 3 As shown, in the simulation, it is assumed that the number of elements in the two subarrays of the sparse nested array are N1 = N2 = 3, then the total number of elements is M = 6. The element spacing of the first subarray is d, and the element spacing of the second subarray is (N1+1)d, where d = λ / 2 is half the wavelength. The number of target radiation sources is K = 2, and their position coordinates are (-500.5, 600.5) and (800.5, 200.5). The number of SNR snapshots for each observation station node starts from 100 and increments to 500 in 100 increments. In this invention, the number of Monte Carlo simulation experiments is MN = 500, and the SNR is 10dB. For this off-grid situation, when the algorithm's positioning performance reaches a bottleneck, Taylor compensation can significantly improve the algorithm's performance. Moreover, when the SNR remains constant, the larger the number of snapshots, the better the estimation performance of the method proposed in this embodiment. Therefore, the positioning accuracy of the DFT-DPD method described in this embodiment can be improved by appropriately increasing the number of snapshots.

[0148] This embodiment receives non-Gaussian signals from K target radiation sources through observation stations, each equipped with a sparse nested array. The fourth-order cumulant is calculated based on the received signals. Spatial smoothing and vectorization of the fourth-order cumulant matrix are then performed to obtain the received signals of continuous virtual array elements. A DFT algorithm spectral function is constructed to estimate the initial position coordinates of the sources. Then, using the array aperture of the vectorized virtual array, Taylor compensation is applied to obtain the Taylor compensation results for the positions of the K radiation sources. The use of fourth-order cumulants contains more information than traditional second-order cumulants and possesses blind Gaussian properties, array expansion characteristics, and phase detectability. Taylor compensation improves resolution and estimation accuracy.

[0149] In addition to the embodiments described above, the present invention may have other implementations. All technical solutions formed by equivalent substitution or equivalent transformation fall within the protection scope claimed by the present invention.

Claims

1. A direct localization method based on fourth-order cumulant DFT Taylor compensation, characterized in that: Includes the following steps: S1. Construct a sparse nested array in the two-dimensional plane xy, and receive non-Gaussian signals from K target radiation sources through the sparse nested arrays equipped at each observation station; S2. Calculate the fourth-order cumulant of the non-Gaussian signal received by the sparse nested array, and obtain... Matrix, pair The matrix is ​​vectorized, sorted, and redundancy removed to obtain... matrix; S3. Construct the DFT algorithm spectrum function to obtain the initial position coordinates of each target radiation source; S4. Using the array aperture of the virtual array, Taylor compensation is performed to obtain the Taylor compensation results for the positions of the K radiation sources.

2. The direct localization method based on fourth-order cumulant DFT Taylor compensation according to claim 1, characterized in that: In step S1, the sparse nested matrix consists of array elements with the following numbers: and It consists of two uniform subarrays, with a total number of array elements. The spacing between the elements of the first subarray is The spacing between the elements of the second subarray is , If the wavelength is half a wavelength, then the positions of all physical array elements are represented as follows: ； The coordinates of the K radiation sources are: The coordinates of the L observation stations are Then the azimuth angle of the k-th radiation source incident on the l-th observation station is , represented as ； At time t, the signals received by the sparse nested array of the l-th observation station from the K target radiation sources are represented as: ； in, Let represent the direction matrix, which is an M×K dimensional matrix composed of the direction vectors of K radiation sources; where the direction vector of the k-th radiation source is . , Indicates wavelength; These represent the positions of each physical array element; Let represent the zero-mean non-Gaussian signal vector from the k-th radiation source incident on the l-th observation station, and satisfy ... , It is the total number of snapshots; This indicates that the mean is zero and the variance is... The additive noise vector, where the noise and signal are independent, is Gaussian white noise. This is a transpose.

3. The direct localization method based on fourth-order cumulant DFT Taylor compensation according to claim 2, characterized in that: In step S2, the signal characteristics are characterized by a fourth-order cumulant. The definition of the fourth-order cumulant for a known zero-mean N-dimensional stationary random process x is as follows: ； in, This indicates the operation of calculating the cumulative amount. Indicates conjugation, therefore the received signal The fourth-order cumulant is calculated by the following formula: ； in, ,visible Depend on constitute, , The fourth-order cumulant is expressed as In order to obtain the received signal of the virtual array, for Virtualization is performed to obtain the virtual received signal from the l-th observation station: ； in, Let T be the single-shot signal obtained from the virtual array, and let T be... The length.

4. The direct localization method based on fourth-order cumulant DFT Taylor compensation according to claim 3, characterized in that: Step S3 includes the following sub-steps. S3.1 Construct a search grid near the radiation source, searching in both the x and y directions. , Next, if the coordinates of the search grid points are (x, y), then for the grid points near the k-th radiation source, calculate the azimuth angle of each grid point relative to the l-th observation station: ； S3.2, the virtual array's first The direction vector of each signal is ,in The direction vector after DFT processing of the signal is Its q-th element is ； From the above formula, it can be seen that when When it is an integer, There is only the first When the element is not zero, When not an integer, only The elements in the neighborhood are not zero, and all other elements are zero, therefore we search for... The coarse estimated position of non-zero elements ,right Perform initial estimation; S3.3, By receiving the virtual signal A DFT transform is performed to obtain the angle estimate. The positions of the K largest peaks of the received signal vector after the DFT transform are denoted as... , Then the initial angle estimation Represented as Thus we obtain: ； Will Substituting into the above formula, we get: ； S3.4 Constructing the normalized DFT matrix Its t-th element is represented as: ； make The data obtained from L observation stations Adding them together, we get: ； Will Substituting into the above formula, we get: ； in, This represents the t-th element of the single snapshot vector z. ;turn up The maximum value is the rough estimate of the location of the k-th radiation source. .

5. The direct localization method based on fourth-order cumulant DFT Taylor compensation according to claim 4, characterized in that: Step S4 includes the following sub-steps. S4.1 The coarse estimation results of the positions of the K radiation sources obtained by searching the spectral function constructed by the DFT algorithm are as follows: Define the coarse estimation vector on the x-axis. Coarse estimate vector on the y-axis After calculating the fourth-order cumulant of the non-Gaussian radiation source, a virtual uniform linear array is obtained, and the positions of its virtual array elements are represented by a set. To indicate: ； Where T is the total number of elements in the virtual array, i.e., the length of vector z, and ; S4.2, Let the equivalent steering matrix of the virtualized l-th observation station be . ,and The equivalent steering vector is: , in, ； and express The v-th element, This represents the x-coordinate of the coarse estimate of the k-th radiation source. The y-coordinate represents the coarse estimate of the k-th radiation source; S4.3, to exist Performing a first-order Taylor series expansion at the given point and neglecting second-order and higher error terms, we obtain: ； in, and These represent the x-coordinates of the radiation source and the coarse estimation result, respectively. and These represent the y-coordinates of the radiation source and the coarse estimation result, respectively. S4.4, Definition , ,and , , , , The virtual received signal of the l-th observation station is rewritten as: ； but ； The above expression can be written as: ； in, , , , ; S4.5, according to Find and divide it into blocks , , , Representation matrix Lines 1 to K, Representation matrix Lines K+1 to 2K, Representation matrix Lines 2K+1 to 3K; S4.6 The Taylor compensation values ​​for the x-axis and y-axis of the radiation source location estimation results are as follows: ； ； The final location results of the K radiation sources are: ； ； in, This represents the coarse estimate vector on the x-axis. This represents a coarse estimate vector on the y-axis.

6. The direct localization method based on fourth-order cumulant DFT Taylor compensation according to claim 1, characterized in that: It also includes a method for proving the effectiveness of steps S1 to S4, specifically: proof is provided through MATLAB simulation analysis, using the root mean square error as the criterion for evaluating performance, defined as follows: ； Where K is the number of target radiation sources, MN is the number of Monte Carlo simulation experiments, and mn represents the mn-th Monte Carlo simulation experiment. and These are the coarse estimates of the x and y coordinates of the k-th radiation source obtained from the mn-th test, respectively. and These are the x and y coordinates of the radiation source, respectively.

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