A direct positioning method for broadband signals based on multi-frequency cost function fusion under unknown attenuation coefficient

Abstract

The present invention discloses a broadband signal direct positioning method based on multi-frequency cost function fusion under unknown attenuation coefficient, including: distributed monitoring nodes receive and sample broadband signals of radiation sources, and transmit the sampled broadband signals to a central site; the central site integrates all received data, performs data preprocessing after segmenting the received data, and obtains narrowband signals corresponding to each frequency in the broadband signal; the covariance matrix of each narrowband signal is characteristically decomposed to obtain its noise subspace, and a new cost function is constructed according to the orthogonal relationship in the multi-band; the positioning area and the search grid size are selected, and a new cost function is adopted, and the gridded spectrum peak search is performed in the selected area in combination with the Lagrange multiplier method when the attenuation coefficient is unknown, and finally the radiation source coordinate estimation value is obtained. The present invention can effectively reduce the time cost required for positioning and effectively improve the estimation accuracy.

Description

Technical Field

[0001] The invention belongs to the technical field of passive positioning, and in particular relates to a broadband signal direct positioning method based on multi-frequency cost function fusion under unknown attenuation coefficient. Background Art

[0002] Positioning technology has played a huge role in many fields such as wireless communication, signal processing, navigation, and target detection. As an important part of positioning technology, passive positioning technology has gradually received widespread attention since the 1940s. Unlike active positioning, to determine the location of the radiation source, passive positioning equipment does not need to send signals, but only needs to receive signals from the radiation source. Therefore, it has a high degree of concealment. In the military, passive positioning is mainly used for the positioning of radars, communication systems, and some aircraft with low signal-to-noise ratios. In the civilian field, passive positioning has been widely used in indoor navigation.

[0003] Early passive positioning was generally implemented using the traditional two-step positioning method: the first step is to estimate the intermediate parameters using the signals received by the observation station. The most commonly used parameters are direction of arrival (DOA), time of arrival (TOA), frequency difference of arrival (FDOA), time difference of arrival (TDOA), received signal strength (RSS), etc.; the second step is to establish the corresponding equation based on the above parameters and the distribution of observation stations. Then, the equation is solved by a certain method to obtain the location of the radiation source.

[0004] Although the traditional two-step positioning method is widely used, it also has its shortcomings. Obviously, the performance of the estimation is limited by the intermediate parameter estimation in the first step. In addition, in the two-step positioning method, the correlation between the signals received by different observation stations from the same radiation source is not considered. As wireless positioning technology is widely used in more complex positioning scenarios, the shortcomings of the two-step positioning method, such as insufficient positioning accuracy and resolution, are gradually becoming apparent.

[0005] In order to avoid the above problems and improve the positioning performance, the direct positioning (DPD) method was proposed. This technology uses multi-array observation stations to fuse the angle information of the target into the array manifold. DPD does not require parameter estimation and can directly estimate the target position from the received data of the observation station. Therefore, the DPD method can significantly improve the positioning performance, especially when the signal-to-noise ratio is low. However, due to the need to perform multi-dimensional matrix operations, the algorithm complexity of DPD is relatively high, the amount of calculation is often large, and it takes a long time. In addition, the array-based DPD method requires a higher hardware cost, while the distributed base station composed of a single antenna has a lower cost and can be used to implement the currently rare TDOA-based DPD method. Summary of the invention

[0006] The technical problem to be solved by the present invention is to address the deficiencies of the above-mentioned prior art and to provide a direct positioning method for broadband signals based on multi-frequency cost function fusion under unknown attenuation coefficients. The method is based on distributed monitoring nodes to receive radiation source signals, utilizes signal segmentation and decomposes broadband signals into a set of narrowband signals, which is beneficial to reducing the amount of matrix calculations, can effectively reduce the time cost required for positioning, and does not require parameter estimation, which can effectively improve the estimation accuracy. In addition, the use of distributed single-antenna observation nodes for positioning helps save costs compared to array antennas.

[0007] In order to achieve the above technical objectives, the technical solution adopted by the present invention is:

[0008] A broadband signal direct positioning method based on multi-frequency cost function fusion under unknown attenuation coefficient, comprising:

[0009] Step 1: The distributed monitoring node receives and samples the broadband signal from the radiation source and transmits the sampled broadband signal to the central site;

[0010] Step 2: The central site integrates all received data, segments the received data, and performs data preprocessing to obtain the narrowband signal corresponding to each frequency in the broadband signal;

[0011] Step 3: Perform eigendecomposition on the covariance matrix of each narrowband signal to obtain its noise subspace, and construct a new cost function based on the orthogonal relationship in multiple frequency bands;

[0012] Step 4: Select the positioning area and search grid size, adopt a new cost function, combine the Lagrange multiplier method when the attenuation coefficient is unknown, perform a gridded spectrum peak search in the selected area, and finally obtain the estimated value of the radiation source coordinates.

[0013] To optimize the above technical solutions, the specific measures taken also include:

[0014] The specific steps of step 1 above include:

[0015] (11) Assume that there are M radiation sources with unknown positions, and their coordinates are: P1(x1,y1,z1),…,P M (x M ,y M ,z M ), the number of distributed monitoring nodes is L, and the coordinates are S1(x s1 ,y s1 ,z s1 ), …, S L (x sL ,y sL ,z sL );

[0016] Assume the sampling rate is fs , then the received signal of each monitoring node is expressed as:

[0017]

[0018] Among them, s m (t) represents the emission signal of the mth radiation source at time t, α lm Indicates the signal attenuation on the propagation path;

[0019] n l (t) is the noise generated when the corresponding node receives the signal, which is considered to be zero-mean Gaussian noise;

[0020] τ lm It is the time delay of the signal from the radiation source to the monitoring station, and its expression is:

[0021]

[0022] Where c represents the speed of electromagnetic wave propagation;

[0023] (12) Sample the received signal, with a sampling interval of T = 1 / f s , let t = n s T, then the discrete form of the received signal It is expressed as:

[0024]

[0025] Among them, N s Indicates the number of sampling points of the signal.

[0026] The specific steps of step 2 above include:

[0027] (21) The central site receives and integrates the signals of all monitoring nodes to obtain a matrix whose row vectors correspond to the signals received by the monitoring nodes. The number of columns in this matrix is ​​N. S , divide it into J sub-matrices evenly by column, then the number of columns in each sub-matrix is ​​N = N S / J, the broadband signal is divided into J segments;

[0028] (22) Perform frequency domain transformation on the row vector of each submatrix, i.e., the received signal after segmentation, to separate the time delay and frequency information in the signal. At this time, the spectrum envelope of the signal before and after segmentation is the same, and each segment of the signal consists of information of N frequency points;

[0029] (23) The spectrum information corresponding to the same frequency of the segmented signals of the same monitoring node is taken out in order to form a narrowband data at this frequency, where the nth signal is X n(j), at this time, the original broadband signal is divided into a set of N narrowband signals of length J, and the matrix dimension of the signal corresponding to each frequency is L×J.

[0030] In (21) above, the signals received by all observation stations are integrated to obtain the composite signal The expression is:

[0031]

[0032] The matrix y(n s ) is evenly divided into J sub-matrices, each of which has N=N columns. s / J, the segmented signal is expressed as

[0033] y(n)=[y1(n) y2(n) … y J (n)],1≤n≤N

[0034] in, In the formula, Among them, s mj (n) = s m (n s ),(j-1)N+1≤n s ≤jN.

[0035] In (22) above, after the data is segmented, j (n) is transformed into the frequency domain, and we get

[0036]

[0037] Among them, Y lj (k) is y lj (n) frequency domain form, and Indicates the corresponding segment in the transmitted signal;

[0038] S mj (k) means s mj DFT of (n), N j (k) is the DFT of the noise after segmentation;

[0039] τ lm is a function of the radiation source position and can be expressed as τ l (p m );

[0040] Among them, p m represents the position of the mth radiation source;

[0041] but Rewritten as:

[0042]

[0043] Among them, the matrix Φ(α) contains all the attenuation coefficients, A(ω k , p) is the delay factor containing position information, and its expressions are:

[0044]

[0045]

[0046] Y j (k) contains N groups of information of different frequencies.

[0047] The above step 3 includes the following steps:

[0048] (31) Calculate the covariance matrix corresponding to all narrowband signals and perform eigendecomposition on them to obtain their noise subspace;

[0049] (32) According to the orthogonal relationship in each frequency band, the corresponding single-band cost function is written, and the cost functions under all frequency bands are integrated to obtain a new cost function for multi-frequency fusion.

[0050] In (31) above, The covariance matrix of is subjected to eigendecomposition to obtain:

[0051]

[0052] in, represents the signal subspace of the nth narrowband signal, represents the noise subspace.

[0053] In (32) above, the orthogonal relationship between the noise subspace and the node and the radiation source direction manifold is obtained:

[0054]

[0055] in, represents the direction vector;

[0056] Represents the vector composed of target coordinates;

[0057] Contains all attenuation coefficients for each observation station;

[0058] Combining the equations for all frequencies into the following:

[0059]

[0060] The new cost function is defined by the above formula:

[0061]

[0062] The above step 4 is specifically implemented as follows:

[0063] The area of ​​interest is divided as the positioning area, and the size of the search grid is defined. The coordinates of each grid point are substituted into the cost function. According to the Lagrange multiplier method, the maximum value of the first element of the cost function is searched in the selected area. The coordinates corresponding to the maximum value are the estimated coordinates of the radiation source.

[0064] The present invention has the following beneficial effects:

[0065] The present invention is applicable to a distributed monitoring node positioning scenario consisting of multiple single-antenna observation stations.

[0066] The specific steps of the present invention are as follows: first, the received signal is segmented and data preprocessing is performed in different frequency bands to obtain noise subspaces in different frequency bands; then, the orthogonal relationship existing in multiple frequency bands of the received signal is used to give the cost function at each frequency, and based on the common attenuation coefficient, the cost function of multi-frequency fusion is further obtained; since the attenuation coefficient of the signal is unknown, the Lagrange multiplier method is combined to perform parameter separation estimation, and the estimation of the radiation source position can finally be obtained through spatial search.

[0067] The present invention uses a single-antenna distributed monitoring node to replace a conventional array antenna, effectively saving the cost; converting a broadband signal into a collection of multi-band narrowband signals, and through data segmentation, greatly reducing the complexity of matrix operations, saving time costs, and improving positioning efficiency; compared with the traditional two-step positioning method, the direct positioning method used in the present invention avoids parameter estimation and reduces positioning errors; unlike most direct positioning methods based on the assumption of a narrowband signal source, the present invention can be applied to the positioning of a single or multiple broadband signal sources.

[0068] The present invention targets broadband signals that are becoming more and more widely used. By performing signal segmentation and frequency band division related processing, the algorithm's computational complexity is effectively reduced. At the same time, compared with traditional time difference positioning and direct positioning, better positioning performance can be obtained. Related aspects have been verified by simulation and actual measurement. In addition, compared with the direct positioning method based on array antennas, the present invention uses a single-antenna monitoring station, which is beneficial to reducing positioning costs. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] Figure 1 A flow chart of a broadband signal direct positioning method based on multi-frequency cost function fusion under unknown attenuation coefficient provided by the present invention;

[0070] Figure 2 This is a schematic diagram of data frequency band division in step 2;

[0071] Figure 3This is a comparison diagram of positioning errors of the method of the present invention and the traditional direct positioning and TDOA methods for simulation data;

[0072] Figure 4 A scene diagram of measured data provided by the present invention;

[0073] Figure 5 It is a thermal map of the positioning area under the measured data by the method of the present invention;

[0074] Figure 6 This is the CDF diagram of the positioning error of the method of the present invention, the traditional direct positioning and the TDOA method for the measured data. DETAILED DESCRIPTION

[0075] The embodiments of the present invention are further described in detail below in conjunction with the accompanying drawings.

[0076] The detailed process of the broadband signal direct positioning method based on multi-frequency cost function fusion under unknown attenuation coefficient provided by the present invention is as follows: Figure 1 As shown:

[0077] The specific implementation is as follows:

[0078] Step 1: The distributed monitoring node receives and samples the broadband signal from the radiation source and transmits the sampled broadband signal to the central site;

[0079] (11) Assume that there are M radiation sources with unknown positions, and their coordinates are: P1(x1,y1,z1),…,P M (x M ,y M ,z M ), the number of distributed monitoring nodes is L, and the coordinates are S1(x s1 ,y s1 ,z s1 ), …, S L (x sL ,y sL ,z sL );

[0080] Assume the sampling rate is f s , then the received signal of each monitoring node is expressed as:

[0081]

[0082] Among them, s m (t) represents the emission signal of the mth radiation source at time t, α lm Indicates the signal attenuation on the propagation path;

[0083] n l (t) is the noise generated when the corresponding node receives the signal, which is considered to be zero-mean Gaussian noise;

[0084] τ lm It is the time delay of the signal from the radiation source to the monitoring station, and its expression is:

[0085]

[0086] Where c represents the speed of electromagnetic wave propagation;

[0087] (12) Sample the received signal, with a sampling interval of T = 1 / f s , let t = n s T, then the discrete form of the received signal It is expressed as:

[0088]

[0089] Among them, N s Indicates the number of sampling points of the signal.

[0090] Step 2: The central site integrates all received data, segments the longer received data, and performs data preprocessing to obtain the narrowband signal corresponding to each frequency in the broadband signal:

[0091] The central site segments the received signal and performs data preprocessing, the goal of which is to convert the broadband signal into a collection of several narrowband signals at different frequencies;

[0092] (21) The central site receives and integrates the signals of all monitoring nodes to obtain a matrix whose row vectors correspond to the signals received by the monitoring nodes. The number of columns in this matrix is ​​N. S , divide it into K sub-matrices evenly by column, then the number of columns in each sub-matrix is ​​N = N S / K, then the long broadband signal is divided into K segments of shorter signals, which is convenient for subsequent calculations;

[0093] Integrate the signals received by all observation stations to obtain a synthetic signal The expression is:

[0094]

[0095] The matrix y(n s ) is evenly divided into J sub-matrices, each of which has N=N columns. s / J, the segmented signal is expressed as

[0096] y(n)=[y1(n) y2(n) … y J (n)],1≤n≤N

[0097] in, In the formula, Among them, smj (n) = s m (n s ),(j-1)N+1≤n s ≤jN;

[0098] (22) Perform frequency domain transformation on the row vector of each submatrix, i.e., the received signal after segmentation, to separate the time delay and frequency information in the signal. At this time, the spectrum envelope of the signal before and after segmentation should be roughly the same, and each segment of the signal consists of information of N frequency points;

[0099] After the data is segmented, j (n) is transformed into the frequency domain, and we get

[0100]

[0101] Among them, Y lj (k) is y lj (n) frequency domain form, and Indicates the corresponding segment in the transmitted signal;

[0102] S mj (k) means s mj DFT of (n), N j (k) is the DFT of the noise after segmentation;

[0103] τ lm is a function of the radiation source position, so it can be expressed as τ l (p m );

[0104] Among them, p m represents the position of the mth radiation source;

[0105] but Rewritten as:

[0106]

[0107] Among them, the matrix Φ(α) contains all the attenuation coefficients, A(ω k , p) is the delay factor containing position information, and its expressions are:

[0108]

[0109]

[0110] Y j (k) contains N groups of information of different frequencies.

[0111] (23) In order to avoid directly processing broadband signals, the spectrum information corresponding to the same frequency of the segmented signals of the same monitoring node is taken out in sequence to form a narrowband data at that frequency. At this time, the original broadband signal is divided into a set of N narrowband signals of length J, and the matrix dimension of the signal corresponding to each frequency is L×J.

[0112] The signal frequency band is divided as follows Figure 2 As shown. Define the nth signal as X j (j). At this time, J becomes the length of the narrowband signal. In reality, the spectrum of the received broadband signal at some frequencies is not ideal. In this case, some frequencies with higher spectrum can be selected as the processed signal.

[0113] Step 3: Perform eigendecomposition on the covariance matrix of each narrowband signal to obtain its noise subspace, and construct a new cost function based on the orthogonal relationship in multiple frequency bands;

[0114] By combining the cost functions at different frequencies, a new cost function for multi-band fusion can be obtained;

[0115] (31) Calculate the covariance matrix corresponding to all narrowband signals and perform eigendecomposition on it to obtain its noise subspace; The covariance matrix of is subjected to eigendecomposition to obtain:

[0116]

[0117] in, represents the signal subspace of the nth narrowband signal, represents the noise subspace.

[0118] (32) According to the orthogonal relationship between the noise subspace and the node and the radiation source direction manifold, we have

[0119]

[0120] in represents the direction vector;

[0121] A vector representing the target coordinates.

[0122] Contains all attenuation coefficients for each observing station.

[0123] Since the unknown attenuation coefficients of narrowband signals of different frequencies of the same broadband transmission signal are the same, that is, they have the same Therefore, the equations for all frequencies can be combined into the following equation:

[0124]

[0125] The new cost function can be defined from the above formula

[0126]

[0127] Step 4: Select the positioning area and search grid size, use the cost function, combine the Lagrange multiplier method when the attenuation coefficient is unknown, perform a gridded spectrum peak search in the selected area, and finally obtain the estimated value of the radiation source coordinates.

[0128] The selected area is gridded, the cost function values ​​of the grid points in the selected area are calculated, and a spectrum peak search is performed to obtain an estimate of the radiation source position coordinates.

[0129] First, according to the relationship between the cost function and the attenuation coefficient, we can get However, since the attenuation coefficient is unknown, the Lagrange multiplier method is used to solve this problem.

[0130] Secondly, the positioning area is selected and gridded, and the position of each point in the grid is substituted into the search cost function.

[0131] Assume unknown The first element of is 1, then by the Lagrange multiplier method, we can find The coordinates of the radiation source are estimated by taking the M elements with the largest absolute values ​​in the first row and the first column of .

[0132] Figure 3 This is a comparison chart of the positioning error CDF of the method of the present invention and the traditional direct positioning and TDOA methods for simulation data. The sampling rate of the simulation signal is f s =125MHz, the time domain width of the LFM simulation signal is T0=0.1ms; the bandwidth B=20MHz, the frequency modulation slope k=B / T0, and 40 simulations are performed under the condition of -15dB signal-to-noise ratio. The simulation data are processed by the method of the present invention and the TDOA method in the traditional direct positioning and two-step positioning respectively. The number of signal segments is 320, the number of DFT points is 32, and 15 frequency bands with larger amplitude values ​​are selected, and the cumulative distribution function CDF is used as the error evaluation index. Figure 3 It can be seen from the figure that the positioning accuracy of the method of the present invention is better than that of the other two positioning methods.

[0133] Figure 4 The measured data scene diagram provided by the present invention, wherein Figures (a) and (b) are physical diagrams of the radiation source and the monitoring node, respectively, and Figure (c) is a distribution diagram of four distributed monitoring nodes and a single radiation source. The measured data source of the present invention is the radiation source signal received by these monitoring nodes.

[0134] Figure 5It is the thermal map of the positioning area under the measured data by the method of the present invention. The sampling rate of the measured data is f s =125MHz, the signal length is 32508, the number of DFT points is 256, and the number of segments is 126. The signals corresponding to the 15 frequencies with larger amplitude values ​​in the spectrum are selected for positioning, and the heat map of the positioning area is obtained. Figure 5 In the figure, the overall area is the selected positioning range of 400m×400m, and the grid side length is 1m. Figure 5 The Target area in represents an area with a higher thermal value, that is, the estimated target position. As can be seen from the figure, for the measured data, the positioning result of the method of the present invention is relatively accurate.

[0135] Figure 6 This is the CDF diagram of the positioning error of the method of the present invention and the traditional direct positioning and TDOA method for measured data. Among them, a positioning range of 200m×200m is selected, the grid side length is 1m, the number of DFT points is 128, and the signals corresponding to the 9 frequencies with larger amplitude values ​​in the spectrum are selected for positioning. The 20 sets of measured data collected are estimated and the error is calculated. Figure 6 In the figure, the horizontal axis represents the error range of the positioning result, and the vertical axis represents the CDF value. Figure 6 It can be seen that the error generated by the method of the present invention is significantly smaller than that of the other two traditional algorithms, which reflects the superiority of the method of the present invention in estimation accuracy and good applicability in actual situations.

[0136] The above are only preferred embodiments of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions under the concept of the present invention belong to the protection scope of the present invention. It should be pointed out that for ordinary technicians in this technical field, some improvements and modifications without departing from the principle of the present invention should be regarded as the protection scope of the present invention.

Claims

1. A broadband signal direct positioning method based on multi-frequency cost function fusion under unknown attenuation coefficient, characterized in that: include: Step 1: The distributed monitoring node receives and samples the broadband signal from the radiation source and transmits the sampled broadband signal to the central site; Step 2: The central site integrates all received data, segments the received data, and performs data preprocessing to obtain the narrowband signal corresponding to each frequency in the broadband signal; Step 3: Perform eigendecomposition on the covariance matrix of each narrowband signal to obtain its noise subspace, and construct a new cost function based on the orthogonal relationship in multiple frequency bands; Step 4: Select the positioning area and search grid size, use the new cost function, combine the Lagrange multiplier method when the attenuation coefficient is unknown, perform a gridded spectrum peak search in the selected area, and finally obtain the estimated value of the radiation source coordinates; The specific steps of step 1 include: (11) Assume that there are M radiation sources with unknown positions, and their coordinates are: P1(x1,y1,z1),…,P M (x M ,y M ,z M ), the number of distributed monitoring nodes is L, and the coordinates are S1(x s1 ,y s1 ,z s1 ), …, S L (x sL ,y sL ,z sL ); Assume the sampling rate is f s , then the received signal of each monitoring node is expressed as: Among them, s m (t) represents the emission signal of the mth radiation source at time t, α lm Indicates the signal attenuation on the propagation path; n l (t) is the noise generated when the corresponding node receives the signal, which is considered to be zero-mean Gaussian noise; τ lm It is the time delay of the signal from the radiation source to the monitoring station, and its expression is: Where c represents the speed of electromagnetic wave propagation; (12) Sample the received signal, with a sampling interval of T = 1 / f s , let t = n s T, then the discrete form of the received signal It is expressed as: Among them, N s Indicates the number of sampling points of the signal; The specific steps of step 2 include: (21) The central site receives and integrates the signals of all monitoring nodes to obtain a matrix whose row vectors correspond to the signals received by the monitoring nodes. The number of columns in this matrix is ​​N. S , divide it into J sub-matrices evenly by column, then the number of columns in each sub-matrix is ​​N = N S / J, the broadband signal is divided into J segments; (22) Perform frequency domain transformation on the row vector of each submatrix, i.e., the received signal after segmentation, to separate the time delay and frequency information in the signal. At this time, the spectrum envelope of the signal before and after segmentation is the same, and each segment of the signal consists of information of N frequency points; (23) The spectrum information corresponding to the same frequency of the segmented signals of the same monitoring node is taken out in order to form a narrowband data at this frequency, where the nth signal is X n (j), the original broadband signal is divided into a set of N narrowband signals of length J, and the matrix dimension of each frequency corresponding signal is L×J; In (21), the signals received by all observation stations are integrated to obtain the synthetic signal The expression is: The matrix y(n s ) is evenly divided into J sub-matrices, each of which has N=N columns. s / J, the segmented signal is expressed as y(n)=[y1(n)y2(n)y J (n)],1≤n≤N in, In the formula, Among them, s mj (n) = s m (n s ),(j-1)N+1≤n s ≤jN; In (22), after the data is segmented, y j (n) is transformed into the frequency domain, and we get Among them, Y lj (k) is y lj (n) frequency domain form, and Indicates the corresponding segment in the transmitted signal; S mj (k) means s mj DFT of (n), N j (k) is the DFT of the noise after segmentation; τ lm is a function of the radiation source position and is denoted by τ l (p m ); Among them, p m represents the position of the mth radiation source; but Rewritten as: Among them, the matrix Φ(α) contains all the attenuation coefficients, A(ω k , p) is the delay factor containing position information, and its expressions are: Y j (k) contains N groups of information of different frequencies; The step 3 includes the following steps: (31) Calculate the covariance matrix corresponding to all narrowband signals and perform eigendecomposition on them to obtain their noise subspace; (32) According to the orthogonal relationship in each frequency band, the corresponding single-band cost function is written, and the cost functions under all frequency bands are integrated to obtain a new cost function of multi-frequency fusion; In (31), The covariance matrix of is subjected to eigendecomposition, and we get: in, represents the signal subspace of the nth narrowband signal, represents the noise subspace; In (32), the orthogonal relationship between the noise subspace and the node and the radiation source direction manifold is obtained: in, represents the direction vector; Represents the vector composed of target coordinates; Contains all attenuation coefficients for each observation station; Combining the equations for all frequencies into the following: The new cost function is defined by the above formula: The step 4 is specifically implemented as follows: The area of ​​interest is divided as the positioning area, and the size of the search grid is defined. The coordinates of each grid point are substituted into the new cost function. According to the Lagrange multiplier method, the maximum value of the first element of the cost function is searched in the selected area. The coordinates corresponding to the maximum value are the estimated coordinate values ​​of the radiation source.

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