A direct positioning method for quasi-stationary signals in nested arrays

Abstract

The present invention discloses a direct positioning method for quasi-stationary signals under a nested array, by arranging a number of base stations carrying nested arrays to receive multiple quasi-stationary signals, using the characteristics of its second-order characteristics being locally stable and changing between different frames, calculating the covariance frame by frame, vectorizing and removing noise to obtain a virtual receiving signal matrix; then using the Khatri‑Rao subspace method to perform dimensionality reduction transformation; then correcting the original dimensionality reduction transformation matrix and using the unitary transformation method to construct a cost function; finally, performing a spectrum peak search on the positioning cost function to complete the radiation source position estimation. The advantage of the present invention is that the quasi-stationary signal characteristics are taken into account in the direct positioning method, and compared with the general direct positioning method, more signal sources can be positioned simultaneously, the positioning error is smaller, and the signal source resolution is higher; the correction of the dimensionality reduction matrix and the application of the unitary transformation greatly reduce the complexity of the method and the calculation cost is lower.

Description

Technical Field

[0001] The present invention relates to the field of passive positioning technology, and in particular to a direct positioning method for quasi-stationary signals under a nested array. Background Art

[0002] In real-world positioning scenarios, in many cases, the radiation source signal is not stable, but remains locally stable for a period of time, and shows differences in different time periods. Such unstable signals are also called quasi-stationary signals, such as audio and speech signals. Currently, the positioning methods for quasi-stationary signals are mostly in the DOA method for the purpose of direction finding, and there is no direct positioning method for such signals.

[0003] Since the direct positioning method does not need to estimate intermediate parameters, it avoids the error transmission and parameter matching problems of the two-step positioning, so it can achieve higher positioning accuracy than the two-step positioning method. In addition, previous studies have shown that introducing the specific characteristics of the radiation source signal in the direct positioning method model can further improve the estimation performance of the method. However, most of these direct positioning methods regard the radiation source signal as a stationary signal, without considering its quasi-stationary characteristics that are more likely to be exhibited in the real world. Summary of the invention

[0004] The technical problem to be solved by the present invention is to provide a direct positioning method for quasi-stationary signals in a nested array in view of the defects involved in the background technology.

[0005] The present invention adopts the following technical solutions to solve the above technical problems:

[0006] A method for directly locating a quasi-stationary signal under a nested array includes the following steps:

[0007] Step 1), using N base stations equipped with the same nested linear array to receive multiple mutually unrelated quasi-stationary signals for sampling;

[0008] Step 2), using the characteristics of quasi-stationary signals, the covariance matrix is ​​calculated frame by frame, vectorized and the unknown noise is removed to construct a virtual received signal matrix;

[0009] Step 3), using the Khatri-Rao subspace method to perform dimensionality reduction transformation on the virtual received signal matrix;

[0010] Step 4), correcting the virtual received signal matrix after the dimensionality reduction transformation to obtain a corrected dimensionality reduction matrix; obtaining a real-valued received signal matrix based on the corrected dimensionality reduction matrix combined with a unitary transformation method, and calculating its corresponding real-valued covariance matrix;

[0011] Step 5), perform eigenvalue decomposition on the real-valued covariance matrix to obtain a real-valued noise subspace, construct a real-valued positioning cost function based on the real-valued noise subspace using the idea of ​​subspace data fusion, and perform spectrum peak search on the positioning cost function to complete the radiation source position estimation.

[0012] As a further optimization scheme of the direct positioning method for quasi-stationary signals under a nested array of the present invention, the N base stations in step 1) are equipped with an array structure configured as M 1 and M 2 , a two-level nested linear array with a total number of array elements of M, and the specific array element positions are

[0013] Θ={d i =id|i=1,2,…,M 1 -1}∪{d j =j(M 1 +1)d+M 1 d|j=1,2,…,M 2 -1}

[0014] Where, d is the unit array element spacing;

[0015] The signal vector received by the nth base station at the tth sampling time is in, Represents about θ n,k The guiding vector, is the angle at which the kth radiation source is incident on the nth base station, q k =[x k ,y k ] is the position of the radiation source, is the location of the base station, K represents the number of radiation sources, d m ∈Θ represents the position of the array element, λ is the signal wavelength, s n,k (t) represents the source signal envelope, n l (t) represents Gaussian white noise vector;

[0016] Received signal matrix X n =A n S n +N n ,in:

[0017]

[0018]

[0019]

[0020]

[0021] represents a set of M×N dimensional complex matrices, and T is the total number of sampling snapshots.

[0022] As a further optimization scheme of the direct positioning method for quasi-stationary signals under a nested array of the present invention, the specific steps of constructing the virtual receiving signal matrix in step 2) are as follows:

[0023] Assume that there are F frames of received signals, each containing T F According to the quasi-stationary characteristics of the source signal, E{|s n,k,f | 2 =g n,k,f , so the signal matrix X received in the fth frame n,f It is expressed as:

[0024] X n,f =[x n ((f-1)T F +1),x n ((f-1)T F +2),…,x n (fT f )]

[0025] =A n S n,f +N n,f

[0026] Among them, S n,f =[s n,1,f ,s n,2,f ,…,s n,K,f ] T represents the source signal matrix of the f-th frame, N n,f For S n,f The corresponding noise matrix;

[0027] Calculate the covariance matrix of the received signal of frame f Among them, Λ n,f =diag{g n,1,f ,g n,2,f ,…,g n,K,f} represents the covariance matrix of the source signal of the frame, C n is the noise covariance matrix;

[0028] Vectorize the covariance matrix and get

[0029]

[0030] in, Because n,f A vector consisting of diagonal elements, represents a set of M×N dimensional real matrices;

[0031] Stack the vectorized forms of all frame signals to construct a new received signal matrix Y n for

[0032]

[0033] in, is the new source signal matrix, 1 F represents an F-dimensional column vector with all 1s;

[0034] To remove unknown noise Defining the orthographic projection matrix Among them I F is an F×F dimensional unit matrix, so the virtual received signal matrix is ​​constructed

[0035]

[0036] As a further optimization scheme of the direct positioning method for quasi-stationary signals under a nested array of the present invention, the specific steps of step 3) are as follows:

[0037] In the Khatri-Rao After that, we get 2R NA -1(R NA =M 2 (M 1 +1)) continuous virtual array elements, take the non-negative part as the virtual uniform linear array, regard the used nested array as a sub-array of the virtual uniform linear array, and their direction matrices satisfy in, To select the matrix, is a real-valued vector whose jth element is 1 and all other elements are 0, j = 1 + d m / d, represents the direction matrix of the virtual linear array, is its kth steering vector;

[0038] but in:

[0039]

[0040] is the virtual direction matrix after Khatri-Rao dimension reduction, is its kth steering vector;

[0041] Defining the Matrix Let the reduced dimension matrix right Perform dimensionality reduction transformation to obtain

[0042] As a further optimization scheme of the direct positioning method for quasi-stationary signals under a nested array of the present invention, the specific steps of step 4) are as follows:

[0043] Pair Matrix Left multiplication So there is

[0044]

[0045] in, is the corrected received signal matrix, is the modified dimension reduction matrix;

[0046] Define the unitary matrix in, and are column vectors of all 1s and all 0s respectively. is the unit matrix, It is a matrix whose anti-diagonal elements are all 1 and other elements are all 0;

[0047] Will Convert to real-valued received signal matrix

[0048]

[0049] Compute real-valued covariance matrix as follows

[0050]

[0051] As a further optimization scheme of the direct positioning method for quasi-stationary signals under a nested array of the present invention, the specific steps of step 5) are as follows:

[0052] right Perform eigenvalue decomposition:

[0053]

[0054] in, represents the real-valued signal subspace consisting of the eigenvectors corresponding to the largest K eigenvalues, represents the real-valued noise subspace consisting of the remaining eigenvectors, They are diagonal matrices composed of the largest K eigenvalues ​​and the remaining eigenvalues ​​arranged in descending order;

[0055] Therefore, the real-valued noise subspace of all base station received signals is integrated to construct the real-valued positioning cost function sp R-QSS-SDF (p) is:

[0056]

[0057] in, is the real-valued search manifold vector; p is the grid point position in the search area Ξ; the search area Ξ is a grid point with a length of O l , O r , O u and O d is a two-dimensional plane region with left, right, top and bottom boundaries;

[0058] Finally, look for sp R-QSS-SDF The coordinates of the K maximum points of (p) are taken as the estimated values ​​of the radiation source position.

[0059] Compared with the prior art, the present invention adopts the above technical solution and has the following technical effects:

[0060] 1. Can estimate more radiation sources simultaneously;

[0061] 2. Has higher positioning accuracy;

[0062] 3. Has higher source resolution. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 A flowchart of a method for directly locating a quasi-stationary signal under a nested array provided by the present invention;

[0064] Figure 2 A diagram of a secondary nested array structure used in the method of the present invention;

[0065] Figure 3 is a scatter plot of the method of the present invention;

[0066] Figure 4 A comparison diagram of positioning errors between the method of the present invention and a general subspace data fusion method when the signal-to-noise ratio changes;

[0067] Figure 5 A comparison diagram of positioning errors between the method of the present invention and a general subspace data fusion method when the number of signal frames changes;

[0068] Figure 6 This is a comparison diagram of the resolution probabilities of the method of the present invention and a general subspace data fusion method when the distance between radiation sources changes. DETAILED DESCRIPTION

[0069] The technical solution of the present invention is further described in detail below in conjunction with the accompanying drawings:

[0070] The present invention can be implemented in many different forms and should not be considered to be limited to the embodiments described herein. On the contrary, these embodiments are provided to make this disclosure thorough and complete, and will fully express the scope of the present invention to those skilled in the art. In the accompanying drawings, components are enlarged for clarity.

[0071] Aiming at the quasi-stationary characteristics of radiation source signals in actual scenarios, the present invention combines the aperture expansion and degree of freedom advantages of nested arrays, proposes a direct positioning method for quasi-stationary signals under nested arrays, vectorizes the covariance matrix and performs denoising transformation to construct a virtual receiving signal matrix, adopts the Khatri-Rao subspace method to perform dimensionality reduction transformation, combines the idea of ​​subspace data fusion to construct a preliminary cost function of the direct positioning method, corrects the original dimensionality reduction transformation matrix and combines the unitary transformation method to obtain the real-valued receiving signal matrix, and on this basis, combines the idea of ​​subspace data fusion again to construct the final cost function of the direct positioning method, and completes the radiation source positioning through spectral peak search.

[0072] like Figure 1 As shown, the present invention discloses a method for directly locating a quasi-stationary signal under a nested array, and the scheme includes the following steps:

[0073] Step 1), using N base stations equipped with the same nested linear array to receive multiple mutually unrelated quasi-stationary signals for sampling;

[0074] The N base stations are equipped with an array structure configured as M 1 and M 2 , a two-level nested linear array with a total number of array elements of M, the specific array element positions are:

[0075] Θ={d i =id|i=1,2,…,M 1 -1}∪{d j =j(M 1 +1)d+M 1 d|j=1,2,…,M 2 -1}

[0076] Where, d is the unit array element spacing;

[0077] The signal vector received by the nth base station at the tth sampling time is in, Represents about θ n,k The guiding vector, is the angle at which the kth radiation source is incident on the nth base station, q k =[x k ,y k ] is the position of the radiation source, is the location of the base station, K represents the number of radiation sources, dm ∈Θ represents the position of the array element, λ is the signal wavelength, s n,k (t) represents the source signal envelope, n l (t) represents Gaussian white noise vector;

[0078] Received signal matrix X n =A n S n +N n ,in:

[0079]

[0080]

[0081]

[0082]

[0083] represents a set of M×N dimensional complex matrices, T is the total number of sampled snapshots;

[0084] Step 2), using the characteristics of quasi-stationary signals, calculate the covariance matrix frame by frame, vectorize and remove unknown noise to construct a virtual received signal matrix:

[0085] Assume that there are F frames of received signals, each containing T F According to the quasi-stationary characteristics of the source signal, E{|s n,k,f | 2 =g n,k,f , so the signal matrix X received in the fth frame n,f It is expressed as:

[0086] X n,f =[x n ((f-1)T F +1),x n ((f-1)T F +2),…,x n (fT f )]

[0087] =A n S n,f +N n,f

[0088] Among them, S n,f =[s n,1,f ,s n,2,f ,…,s n,K,f ] T represents the source signal matrix of the f-th frame, N n,f For S n,f The corresponding noise matrix;

[0089] Calculate the covariance matrix of the received signal of frame f Among them, Λ n,f =diag{g n,1,f ,g n,2,f ,…,g n,K,f} represents the covariance matrix of the source signal of the frame, C n is the noise covariance matrix;

[0090] Vectorize the covariance matrix and get

[0091]

[0092] in, Because n,f A vector consisting of diagonal elements, represents a set of M×N dimensional real matrices;

[0093] Stack the vectorized forms of all frame signals to construct a new received signal matrix Y n for

[0094]

[0095] in, is the new source signal matrix, 1 F represents an F-dimensional column vector with all 1s;

[0096] To remove unknown noise Defining the orthographic projection matrix Among them I F is an F×F dimensional unit matrix, so the virtual received signal matrix is ​​constructed:

[0097]

[0098] Step 3), using the Khatri-Rao subspace method to perform dimensionality reduction transformation on the virtual received signal matrix;

[0099] In the Khatri-Rao After that, we get 2R NA -1(R NA =M 2 (M 1 +1)) continuous virtual array elements, take the non-negative part as the virtual uniform linear array, regard the used nested array as a sub-array of the virtual uniform linear array, and their direction matrices satisfy in, To select the matrix, is a real-valued vector whose jth element is 1 and all other elements are 0, j = 1 + d m / d, represents the direction matrix of the virtual linear array, is its kth steering vector;

[0100] but in:

[0101]

[0102] is the virtual direction matrix after Khatri-Rao dimension reduction, is its kth steering vector;

[0103] Defining the Matrix Let the reduced dimension matrix right Perform dimensionality reduction transformation to obtain

[0104] Step 4), the virtual received signal matrix after the dimensionality reduction transformation is corrected to obtain a corrected dimensionality reduction matrix; based on the corrected dimensionality reduction matrix combined with the unitary transformation method, the real-valued received signal matrix is ​​obtained, and its corresponding real-valued covariance matrix is ​​calculated:

[0105] Pair Matrix Left multiplication So there is

[0106]

[0107] in, is the corrected received signal matrix, is the modified dimension reduction matrix;

[0108] Define the unitary matrix in, and are column vectors of all 1s and all 0s respectively. is the unit matrix, It is a matrix whose anti-diagonal elements are all 1 and other elements are all 0;

[0109] Will Convert to real-valued received signal matrix

[0110]

[0111] Compute real-valued covariance matrix as follows

[0112]

[0113] Step 5), perform eigenvalue decomposition on the real-valued covariance matrix to obtain a real-valued noise subspace, construct a positioning cost function based on the real-valued noise subspace using the idea of ​​subspace data fusion, and perform spectrum peak search on the positioning cost function to complete the radiation source position estimation:

[0114] right Perform eigenvalue decomposition:

[0115]

[0116] in, represents the real-valued signal subspace consisting of the eigenvectors corresponding to the largest K eigenvalues, represents the real-valued noise subspace consisting of the remaining eigenvectors, They are diagonal matrices composed of the largest K eigenvalues ​​and the remaining eigenvalues ​​arranged in descending order;

[0117] Therefore, the real-valued noise subspace of all base station received signals is integrated to construct the positioning cost function sp R-QSS-SDF (p) is:

[0118]

[0119] in, is the real-valued search manifold vector; p is the grid point position in the search area Ξ; the search area Ξ is a grid point with a length of O l , O r , O u and O d is a two-dimensional plane region with left, right, top and bottom boundaries;

[0120] Finally, look for sp R-QSS-SDF The coordinates of the K maximum points of (p) are taken as the estimated values ​​of the radiation source position.

[0121] To prove that the algorithm of the present invention is superior to the existing algorithm, the following is proved by MATLAB simulation analysis, using the root mean square error (RMSE) and the resolution probability P r As the criteria for evaluating the positioning accuracy and resolution of the method, RMSE and P are defined respectively. r as follows:

[0122]

[0123]

[0124] In the formula, and [x k ,y k ] are the estimated value and true position of the radiation source in the ne-th simulation, respectively. E Indicates the number of Monte Carlo simulations, Ns In the present invention, two radiation sources are placed parallel to the x-axis, one of which is located at q 1 =[x 1 ,y 1 ] (unit: m, the same below), the other is located at q 2 =[x 1 +Δx,y 2 ], Δx represents the distance between the two radiation sources. For each simulation experiment, if the radiation source position estimate and satisfy It is recorded as successful identification, otherwise it is recorded as failure.

[0125] Figure 3 The simulation parameters are set as follows: the number of radiation sources K = 7, located at q 1 =[-800,400],q 2 =[-200,-600],q 3 =[-400,1000],q 4 =[700,1300],q 5 =[1200,0],q 6 =[1600,400],q 7 = [800,500], the number of base stations N = 4, located at v 1 =[0,2000],v 2 =[-700,-1600],v 3 =[1100,-1200],v 4 =[2000,2100], the secondary nested array structure is M 1 =2, M 2 =2, signal-to-noise ratio is 5dB, signal frame number F = 200, number of snapshots per frame T F =600, number of simulation experiments N E = 500. As can be seen from the figure, even if the number of radiation sources exceeds the maximum number of estimable signal sources of the two-level nested array based on the spatial smoothing method (K>R NA -1), the method of the present invention can still complete the positioning relatively accurately, which shows that compared with the general direct positioning method, the method of the present invention can estimate more radiation sources at the same time.

[0126] Figure 4 This is a comparison diagram of the positioning error between the method of the present invention and the general subspace data fusion method when the signal-to-noise ratio changes. The simulation parameters are set as follows: the number of radiation sources K = 2, q 1 =[0,390],q 2 = [60,580], number of base stations N = 3, v 1=[-700,-300], v 2 =[200,-500], v 3 =[600,-200], the structure of the uniform linear array is M=4, and the structure of the secondary nested array is M 1 =2, M 2 =2, signal frame number F = 400, number of snapshots per frame T F =300, the signal-to-noise ratio range is -3dB to 15dB, and the number of simulation experiments is N E =1000. Figure 4 The results show that no matter whether a uniform linear array or a nested array is used, the positioning accuracy of the method of the present invention is higher than that of the general subspace data fusion method and is closest to the Cramer-Rao bound.

[0127] Figure 5 This is a comparison chart of the positioning errors of the method of the present invention and the general subspace data fusion method when the number of signal frames changes. In this simulation, except that the signal-to-noise ratio is set to 5dB and the signal frame number change range is 50 to 400, other parameters are the same as Figure 4 .from Figure 5 The simulation results show that the increase in the number of signal frames has a benefit on the performance improvement of all methods, and the positioning error of the method described in the present invention is still smaller than that of the general subspace data fusion method.

[0128] Figure 6 This is a comparison chart of the resolution probability of the method of the present invention and the general subspace data fusion method when the distance between the radiation sources changes. In this simulation experiment, the signal-to-noise ratio is set to 0dB, the number of signal frames F = 200, and the number of snapshots per frame T F =600, radiation source 1 is located at q 1 = [0,520], another radiation source is located at q 2 =[Δx,520], where Δx varies from 30m to 150m, and other parameters are the same Figure 4 . Figure 6 The results show that when the method of the present invention is applied to nested arrays, Δx is only 50m resolution probability P r It can reach 100%, while the general subspace data fusion method under the same array configuration has a P r This verifies that the method of the present invention has a significant advantage in terms of source resolution compared to the general subspace data fusion method.

[0129] It will be understood by those skilled in the art that, unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as those generally understood by those skilled in the art in the art to which the present invention belongs. It should also be understood that terms such as those defined in common dictionaries should be understood to have meanings consistent with the meanings in the context of the prior art, and will not be interpreted with idealized or overly formal meanings unless defined as herein.

[0130] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for direct positioning of quasi-stationary signals under nested arrays, characterized in that: The following steps are involved: Step 1), using N base stations equipped with the same nested linear array to receive multiple mutually unrelated quasi-stationary signals for sampling; The N base stations are equipped with a two-level nested linear array with array structure configurations of M1 and M2 and a total number of array elements of M. The specific array element positions are Θ={d i =id|i=1,2,…,M1-1}∪{d j =j(M1+1)d+M1d|j=1,2,…,M2-1} Where, .d. is the unit array element spacing; The signal vector received by the nth base station at the tth sampling time is in, Represents about θ n,k The guiding vector, is the angle at which the kth radiation source is incident on the nth base station, q k =[x k ,y k ] is the position of the radiation source, is the location of the base station, K represents the number of radiation sources, d m ∈Θ represents the position of the array element, λ is the signal wavelength, s n,k (t) represents the source signal envelope, n l (t) represents Gaussian white noise vector; Received signal matrix X n =A n S n +N n ,in: represents a set of M×N dimensional complex matrices, T is the total number of sampled snapshots; Step 2), using the characteristics of quasi-stationary signals, the covariance matrix is ​​calculated frame by frame, vectorized and the unknown noise is removed to construct a virtual received signal matrix; Assume that there are F frames of received signals, each containing T F According to the quasi-stationary characteristics of the source signal, E{s n,k,f 2 } = g n,k,f , so the signal matrix X received in the fth frame n,f It is expressed as: X n,f =[x n ((f-1)T F +1),x n ((f-1)T F +2),…,x n (fT f )] =A n S n,f +N n,f Among them, S n,f =[s n,1,f ,s n,2,f ,…,s n,K,f ] T represents the source signal matrix of the f-th frame, N n,f For S n,f The corresponding noise matrix; Calculate the covariance matrix of the received signal of frame f Among them, Λ n,f =diag{g n,1,f ,g n,2,f ,…,g n,K,f } represents the covariance matrix of the source signal of the frame, and Cn is the noise covariance matrix; Vectorize the covariance matrix and get in, Because n,f A vector consisting of diagonal elements, represents a set of M×N dimensional real matrices; Stack the vectorized forms of all frame signals to construct a new received signal matrix Y n for in, is the new source signal matrix, 1 F represents an F-dimensional column vector with all 1s; To remove unknown noise Defining the orthographic projection matrix Among them I F is an F×F dimensional unit matrix, so the virtual received signal matrix is ​​constructed Step 3), using the Khatri-Rao subspace method to perform dimensionality reduction transformation on the virtual received signal matrix; In the Khatri-Rao After that, we get 2R NA -1(R NA =M2(M1+1)) continuous virtual array elements, take the non-negative part as the virtual uniform linear array, regard the used nested array as a sub-array of the virtual uniform linear array, and their direction matrices satisfy in, To select the matrix, is a real-valued vector whose jth element is 1 and all other elements are 0, j = 1 + d m / d, represents the direction matrix of the virtual uniform linear array, is its kth steering vector; but in: is the virtual direction matrix after Khatri-Rao dimension reduction, is its kth steering vector; Defining the Matrix Let the reduced dimension matrix right Perform dimensionality reduction transformation to obtain Step 4), correcting the virtual received signal matrix after the dimensionality reduction transformation to obtain a corrected dimensionality reduction matrix; obtaining a real-valued received signal matrix based on the corrected dimensionality reduction matrix combined with a unitary transformation method, and calculating its corresponding real-valued covariance matrix; Pair Matrix Left multiplication So there is in, is the corrected received signal matrix, is the modified dimension reduction matrix; Define the unitary matrix in, and are column vectors of all 1s and all 0s respectively. is the unit matrix, It is a matrix whose anti-diagonal elements are all 1 and other elements are all 0; Will Convert to real-valued received signal matrix Compute real-valued covariance matrix as follows Step 5), performing eigenvalue decomposition on the real-valued covariance matrix to obtain a real-valued noise subspace, constructing a positioning cost function based on the real-valued noise subspace using the idea of ​​subspace data fusion, and performing spectrum peak search on the positioning cost function to complete the radiation source position estimation; right Perform eigenvalue decomposition: in, represents the real-valued signal subspace consisting of the eigenvectors corresponding to the largest K eigenvalues, represents the real-valued noise subspace consisting of the remaining eigenvectors, They are diagonal matrices composed of the largest K eigenvalues ​​and the remaining eigenvalues ​​arranged in descending order; Therefore, the real-valued noise subspace of all base station received signals is integrated to construct the positioning cost function sp R-QSS-SDF (p) is: in, is the real-valued search manifold vector; p is the grid point position in the search area Ξ; the search area Ξ is a grid point with a length of O l , O r , O u and O d is a two-dimensional plane region with left, right, top and bottom boundaries; Finally, look for sp R-QSS-SDF The coordinates of the K maximum points of (p) are taken as the estimated values ​​of the radiation source position.

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