High-precision positioning method based on angle information fusion estimation

Abstract

The application relates to a high-precision positioning method based on angle information fusion estimation, and belongs to the technical field of wireless positioning algorithms, which comprises the following steps: establishing a benchmark antenna coordinate model; establishing an IQ data frame model according to a CTE protocol model; applying time filtering to the IQ data frame model in the time domain; constructing a mean reference phase difference and an angle phase difference matrix for the IQ data frame model after time filtering processing, and constructing a multi-phase difference composite signal model; constructing a signal covariance matrix by using the multi-phase difference composite signal model, applying spatial smoothing filtering to the signal covariance matrix; and fusing a dimension-reduced signal steering vector obtained by applying spatial smoothing filtering to the signal covariance matrix and a dimension-reduced noise subspace into angle estimation. The application starts from a two-dimensional antenna array spatial coordinate and an IQ data frame model, angle fusion estimation is carried out by using a hierarchical method, and the technical problem of low positioning precision of a two-dimensional antenna array in an indoor environment is solved.

Description

TECHNICAL FIELD

[0001] The application relates to a high-precision positioning method based on angle information fusion estimation, and belongs to the technical field of wireless positioning algorithms. BACKGROUND

[0002] Wireless positioning technology is divided into outdoor wireless positioning technology and indoor wireless positioning technology. In an outdoor environment, a global navigation satellite system (GNSS) such as a global positioning system (GPS) or a Beidou positioning system (BDS) can provide a user with a meter-level positioning service, and can basically solve the problem of accurate positioning in an outdoor space. However, due to the shielding of buildings and other problems, the outdoor wireless positioning technology is not suitable for indoor positioning. In an indoor environment, due to the shielding of obstacles and other problems, the accuracy and robustness of the indoor wireless positioning technology need to be improved.

[0003] There is an urgent need for an indoor positioning system in the industrial field, such as the need to achieve accurate positioning of a trolley and accurate positioning of unmanned cargo loading and unloading in the steel logistics field. Current indoor positioning methods mainly include energy radio frequency-based positioning, time of arrival (TOA)-based positioning, angle of arrival (AOA)-based positioning and the like. The energy radio frequency-based positioning has the advantage of small size, but high-precision positioning needs to deploy dense positioning base stations in advance. The TOA-based positioning has the advantage of high positioning accuracy, but requires a high-precision synchronous clock for the hardware device, and the positioning device has high power consumption. The AOA-based positioning has the advantages of simple positioning method, low power consumption and no need for a high-precision synchronous clock, but there is a technical difficulty that a two-dimensional antenna array is interfered with by electromagnetic interference in an indoor environment, resulting in low positioning accuracy.

[0004] Therefore, there is an urgent need for a technical solution that can solve the problem of low positioning accuracy of a two-dimensional antenna array caused by electromagnetic interference in an indoor environment, to meet the accuracy and robustness requirements of an indoor positioning system in the industrial field. SUMMARY

[0005] The purpose of the present application is to provide a high-precision positioning method based on angle information fusion estimation, which can effectively solve the technical problem of low positioning accuracy of a two-dimensional antenna array caused by electromagnetic interference in an indoor environment.

[0006] To achieve the above purpose, the technical solution adopted by the present application is:

[0007] A high-precision positioning method based on angle information fusion estimation, comprising the following steps:

[0008] Step 1. Establish a reference antenna coordinate model;

[0009] Step 2. Under the reference antenna coordinate model, establish an IQ data frame model with reference to a CTE protocol model;

[0010] Step 3. Time filtering is applied to the IQ data frame model in the time domain;

[0011] Step 4. The mean reference phase difference matrix and the angle phase difference matrix are sequentially constructed based on the time-filtered IQ data frame model, and a multi-phase difference composite signal model is finally constructed;

[0012] Step 5. The signal covariance matrix is constructed based on the multi-phase difference composite signal model, and spatial smoothing filtering is applied to the signal covariance matrix;

[0013] Step 6. The reduced dimension signal steering vector obtained in step 5 is fused with the reduced dimension noise subspace in angle estimation.

[0014] The further improvement of the technical scheme of the present application is that the step 1 is specifically:

[0015] Suppose that there are MxN antennas on a two-dimensional rectangular antenna array, where M represents the number of antenna columns and N represents the number of antenna rows. And suppose that there are K sources, set the upper right corner antenna as the reference antenna, and number the reference antenna as 1, and number the remaining antennas in the order of 2, 3, …, MxN-1, and number the first K sources as 1, 2, …, K.

[0016] The further improvement of the technical scheme of the present application is that the IQ data frame model of the step 2 divides the data into reference IQ data and angle IQ data; the reference IQ data is all collected on the reference antenna 1, and is expressed by the following formula:

[0017]

[0018] Wherein, represents the reference data matrix collected in the sampling period T. represents the I value and the Q value collected in the reference sampling period T.

[0019] The angle IQ data is expressed by the following formula:

[0020]

[0021] Wherein,​​​​​​​​​​​​​​ denotes the angle data matrix under the antenna snapshot period, the antenna snapshot period refers to the time spent for collecting all antennas once in turn, denotes the i-th antenna snapshot. denotes the I value and the Q value under the angle sampling period denotes the antenna numbered denotes the i-th sampling protocol selected.

[0022] in the sampling period The complete IQ data frame model formula is shown as follows:

[0023] .

[0024] Further improvement of the technical scheme of the present application is that the step 3 is specifically:

[0025] The time filtering is applied to the reference IQ data and the angle IQ data respectively, and the filtered IQ data frame model is shown by the following formula:

[0026]

[0027] wherein, denotes the reference IQ data after the time filtering, and the formula is:

[0028]

[0029] is the angle IQ data after the time filtering, and the formula is shown as follows:

[0030] .

[0031] Further improvement of the technical scheme of the present application is that the step 4 is specifically:

[0032] The phase solving calculation formula is applied to the IQ data frame model after the time filtering processing, so as to obtain a phase matrix;

[0033] The phase solving calculation formula is:

[0034]

[0035] wherein denotes an inverse tangent function.

[0036] The phase matrix is:

[0037]

[0038] ​​wherein, is a reference phase matrix calculated by is an angle phase matrix calculated by

[0039] According to the reference phase matrix, the reference phase difference is calculated by using the reference phase difference calculation formula:

[0040]

[0041] wherein, p represents the serial number of the reference phase difference data;

[0042] The data normalization calculation formula and the reference phase difference calculation formula are applied to the reference phase matrix, so as to limit the reference phase difference data in the range of The calculated reference phase difference matrix is represented as follows:

[0043]

[0044] The mean reference phase difference is calculated by using the mean reference phase difference method, and the formula is as follows:

[0045]

[0046] The angle phase and the reference antenna angle phase are differentially calculated, and the mean reference phase difference is applied, to obtain the calculation formula of the mean reference phase difference and the angle phase difference compound application, and then the data normalization calculation formula is applied to the calculation formula. The mean reference phase difference of the reference antenna is used as the first element of the angle phase difference matrix, and the calculation formula of the compound application of the angle phase difference matrix is as follows:

[0047]

[0048] wherein, is an angle phase difference matrix, is an angle phase of any antenna, is a reference antenna angle phase, is a mean reference phase difference, is a sampling switching time, represents a set of antenna serial numbers; Δφ * is a phase difference normalization processing operator; the angle phase difference matrix is calculated as follows:

[0049]

[0050] A multi-phase difference compound signal model is constructed, and is represented as follows:

[0051] ​​

[0052] The further improvement of the technical solution of the present application is that the step 5 is specifically:

[0053] The spatial smoothing filter is applied to the polyphase difference composite signal model, and the filtering result is represented by , and the dimension of the signal covariance matrix is reduced;

[0054] The eigenvalue decomposition is performed on to obtain the reduced dimension signal subspace and the reduced dimension noise subspace, which are represented by the following formulas:

[0055]

[0056] wherein, represents the reduced dimension signal subspace, which is composed of the eigenvectors corresponding to the largest eigenvalues

[0057] represents the reduced dimension noise subspace, which is composed of the eigenvectors corresponding to the remaining small eigenvalues; is composed of the eigenvalues, is composed of the remaining small eigenvalues.

[0058] The reduced dimension signal steering vector is represented by the following formula:

[0059]

[0060]

[0061] wherein, represents the signal steering vector in the axis direction, represents the signal steering vector in the axis direction, and represents the azimuth angle, represents the elevation angle, represents the antenna spacing, , represents the size of the overlapping subarray, represents the signal wavelength; j is the imaginary unit in the complex number field.

[0062] The formula of the signal covariance matrix is:

[0063]

[0064] wherein, represents the mathematical expectation, represents the conjugate transpose of the polyphase difference composite signal model. ​

[0065] Further improvement of the technical scheme of the present application is that the step 6 is specifically:

[0066] The dimension-reduced signal vector obtained in step 5 is fused into angle estimation together with the dimension-reduced noise subspace, and the formula is as follows:

[0067]

[0068] Wherein, represents a spatial spectrum function, represents a Kronecker product;

[0069] The spatial spectrum function is normalized and logarithmized, and the formula is as follows:

[0070]

[0071] Wherein, represents a normalized and logarithmized spatial spectrum function, represents a maximum spectrum peak value in the spatial spectrum function; when estimating the source angle, the azimuth and the pitch angle corresponding to the maximum spectrum peak are found, that is, the source angle estimation value.

[0072] Due to the adoption of the above technical scheme, the present application has the following technical effects:

[0073] The present application sets a reference antenna in a two-dimensional antenna, and applies a space-time joint filtering model and a multi-phase difference complex signal model, and finally establishes a signal matrix and fuses it into angle estimation. The present application takes the two-dimensional antenna spatial coordinates and IQ data as the starting point, uses a hierarchical structure to establish a signal matrix and perform angle estimation fusion, and can solve the problem of low positioning accuracy of a two-dimensional antenna array caused by electromagnetic interference in an indoor environment. BRIEF DESCRIPTION OF DRAWINGS

[0074] Figure 1 is the overall method block diagram of the present application;

[0075] Figure 2 is a reference antenna coordinate three-dimensional model schematic diagram of the present application;

[0076] Figure 3 is a CTE protocol model of the present application. DETAILED DESCRIPTION

[0077] The present application will be further described in detail below in combination with the drawings and specific embodiments:

[0078] A high-precision positioning method based on angle information fusion estimation, as shown in Figure 1 ​As shown, it comprises a data acquisition layer, a data processing layer and an application layer. In the data acquisition layer, first, a reference antenna is set in a two-dimensional antenna, for example, a reference antenna is set in a two-dimensional rectangular antenna, and a corresponding Cartesian coordinate system is established, so as to construct a reference antenna coordinate model. Under the reference antenna coordinate model, a constant tone extension (CTE) protocol model is referred to for establishing an in-phase and quadrature (IQ) data frame model of the signal, and the IQ data frame model divides data into reference IQ data and angle IQ data. In the data processing layer, first, time filtering is applied to the IQ data frame model in the time domain, so as to obtain a more stable IQ data set; then, a multi-phase difference composite signal model is applied to the IQ data frame model processed by time filtering, and the construction of the multi-phase difference composite signal model comprises three processes of constructing a mean reference phase difference, constructing an angle phase difference matrix and constructing a multi-phase difference composite signal model. That is, the mean reference phase difference is obtained by constructing a reference phase difference matrix and normalizing and averaging the matrix; in the process of constructing the angle phase difference matrix, the mean reference phase difference and angle phase difference data are applied to construct the multi-phase difference composite signal model. Finally, spatial smoothing filtering is applied to the signal covariance matrix constructed by the multi-phase difference composite signal model in the space, so as to obtain a reduced dimension signal steering vector and a reduced dimension noise subspace. In the application layer, the reduced dimension signal steering vector and the reduced dimension noise subspace obtained by applying spatial smoothing filtering to the signal covariance matrix constructed by the multi-phase difference composite signal model are fused in angle estimation, for example, in two-dimensional multiple signal classification (2D-MUSIC) angle estimation.

[0079] The specific steps are as follows:

[0080] Step 1. Establishing a reference antenna coordinate model

[0081] Suppose there are N antennas on a two-dimensional rectangular antenna array, where N = M x L, M and L represent the number of antenna columns and the number of antenna rows, respectively. Suppose there are K sources, where K < N. Suppose there are K sources, where K < N. Suppose there are K sources, where K < N. Suppose there are K sources, where K < N. Suppose there are K sources, where K < N. Suppose there are K sources, where K < N. Suppose there are K sources, where K < N. Suppose there are K sources, where K < N. Suppose there are K sources, where K < N. Suppose there are K sources, where K < N. Figure 2 Suppose there are K sources, where K < N.

[0082] Step 2. Under the reference antenna coordinate model, an IQ data frame model is established by referring to a CTE protocol model

[0083] The IQ data frame model divides data into reference IQ data and angle IQ data. For the CTE protocol model, as shown in Figure 3 , the angle of arrival (AOA) based positioning includes two sampling protocol mechanisms with sampling switch time of us and us, respectively. The sampling switch time refers to the time of alternation of switch slot and sampling slot. Both of the two sampling protocol mechanisms include a protection period and a reference period in the early stage. The us before the protection period is called the protection period, and the data sampled in the protection period is discarded; the us after the protection period is called the reference period, and the data sampled in the reference period is reserved. After the reference period, there is a series of alternation of switch slot and sampling slot, in which the data of the switch slot is discarded, and the data of the sampling slot is reserved.

[0084] The IQ data frame model divides data into reference IQ data and angle IQ data when referring to the CTE protocol model. The reference IQ data is collected on the reference antenna

[0085] , and is expressed by the following formula:

[0086]

[0087] wherein, refers to the reference data matrix collected in the sampling period . refers to the I value and the Q value collected in the reference sampling period , and refers to the number of reference IQ data group samples.

[0088] The angle IQ data is expressed by the following formula:

[0089]

[0090] wherein, refers to the angle data matrix in the antenna snapshot period , and the antenna snapshot period refers to the time for collecting all antennas once, refers to the number of antenna snapshots. refers to the I value and the Q value in the angle sampling period , and refers to the antenna numbered , and refers to the selected sampling protocol.

[0091] Since there is a certain phase deviation in the sampling of the same antenna, in order to reduce the deviation, the sampling is performed times. In the sampling period​ The complete IQ data frame model formula is shown as follows:

[0092]

[0093] wherein, represents the complete IQ data frame model at the sampling period , represents the reference IQ data at the sampling period , represents the angle IQ data sampled at the th antenna snapshot.

[0094] Step 3. In order to reduce the noise pollution of the environment to the channel, time filtering is applied to the IQ data frame model in the time domain, that is, time filtering is applied to the reference IQ data and the angle IQ data respectively. The filtered IQ data frame model is represented by the following formula:

[0095]

[0096] wherein, represents the complete IQ data frame model after applying time filtering, represents the sampling

[0097] period. represents the reference IQ data after applying time filtering, and the formula is as follows:

[0098]

[0099] is the angle IQ data after applying time filtering, and the formula is shown as follows:

[0100]

[0101] Step 4. The mean reference phase difference, angle phase difference matrix are constructed in turn based on the IQ data frame model after time filtering, and finally a multi-phase difference composite signal model is constructed

[0102] The phase solving calculation formula is applied to the IQ data frame model after time filtering to obtain a phase matrix.

[0103] The phase solving calculation formula is as follows:

[0104]

[0105] wherein represents an inverse tangent function.

[0106] The phase matrix is as follows:

[0107]

[0108] wherein, is calculated from a reference phase matrix, is calculated from an angular phase matrix;

[0109] wherein, the reference phase matrix is expressed by the following equation:

[0110]

[0111] wherein, denotes a phase calculated from a phase difference.

[0112] The angular phase matrix is expressed by the following equation:

[0113]

[0114] wherein, denotes a phase calculated from a phase difference.

[0115] From the reference phase matrix, a reference phase difference calculation equation is calculated by:

[0116]

[0117] wherein, p denotes a serial number of the reference phase difference data.

[0118] A data normalization calculation equation is expressed by:

[0119]

[0120] wherein, denotes a phase difference.

[0121] By applying the data normalization calculation equation and the reference phase difference calculation equation to the reference phase matrix, the reference phase difference data is limited between and the calculated reference phase difference matrix is expressed by:

[0122]

[0123] A mean reference phase difference is calculated by a reference phase difference mean method and the equation is as follows:

[0124]

[0125] The angle phase is differentially calculated with the reference antenna angle phase, and the mean reference phase difference is applied, to obtain a calculation formula of the mean reference phase difference and the angle phase difference, and then the calculation formula is applied to data normalization calculation formula. In order to make up for the dimension of the angle phase difference matrix, the mean reference phase difference of the reference antenna is used as the first element of the angle phase difference matrix, and the calculation formula of the applied angle phase difference matrix is as follows:

[0126]

[0127] Wherein, is the angle phase difference matrix, is the angle phase of any antenna, is the reference antenna angle phase, is the mean reference phase difference, is the sampling switching time, represents the set of antenna sequence number; Δφ * is the phase difference normalization processing operator.

[0128] The angle phase difference matrix is obtained by calculation:

[0129]

[0130] The multi-phase difference composite signal model is constructed, and is represented as follows:

[0131]

[0132] Step 5. The signal covariance matrix is constructed by using the multi-phase difference composite signal model, and the signal covariance matrix formula is represented as follows:

[0133]

[0134] Wherein, represents the mathematical expectation, represents the conjugate transpose of the multi-phase difference composite signal model.

[0135] In view of the problem that the coherent signal existing in space will confuse the estimation result, the spatial smoothing filter is applied to the multi-phase difference composite signal model, and the filter result is represented by , and the dimension of the signal covariance matrix is reduced.

[0136] The eigenvalue decomposition is performed on to obtain the reduced dimension signal subspace and the reduced dimension noise subspace, and the formula is represented as follows:

[0137]

[0138] Wherein, This refers to the reduced-dimensional signal subspace, which is composed of... The eigenvectors corresponding to the largest eigenvalues

[0139] composition; This refers to the reduced-dimensional noise subspace, which is composed of the remaining... The eigenvectors are composed of the eigenvalues ​​corresponding to the small eigenvalues; It is by It consists of 1 eigenvalues. It is the remainder It consists of a number of small eigenvalues.

[0140] The dimension-reduced signal steering vector is represented by the following formula:

[0141]

[0142]

[0143] in, express Signal steering vector in the axial direction, express The signal steering vector in the axial direction, and Indicates azimuth. Indicates the pitch angle. Indicates the antenna spacing. , Indicates the size of the overlapping subarray. The wavelength of the signal is represented by j, which is the imaginary unit in the complex field. j is defined as j 2 =-1.

[0144] Step 6. Integrate the reduced-dimensional signal steering vector obtained in Step 5 with the reduced-dimensional noise subspace into the angle estimation, for example, into the 2D-MUSIC angle estimation, as shown in the following formula:

[0145]

[0146] in, It represents a spatial spectral function. It represents the Kronecker product.

[0147] To normalize the amplitude of the spatial spectral function, the spatial spectral function is normalized and logarithmically transformed, as shown in the following formula:

[0148]

[0149] in, This represents the spatial spectral function after normalization and logarithmic transformation. The maximum spectral peak value in the spatial spectral function. When estimating the source angle, this is achieved by finding... The maximum spectral peak corresponds to the azimuth angle and the elevation angle of the source direction, i.e. the source angle estimation value.

[0150] The above detailed description according to the present application can be modified by those skilled in the art, or some technical features can be replaced by equivalent ones, as long as the modifications or replacements are within the spirit and principle of the present application.

Claims

1. A high-precision positioning method based on angle information fusion estimation, characterized in that, The method comprises the following steps: Step 1. Establishing a reference antenna coordinate model; Step 2. Under the reference antenna coordinate model, establishing an IQ data frame model with reference to a CTE protocol model; Step 3. Applying time filtering to the IQ data frame model in the time domain; Step 4. Constructing a mean reference phase difference, an angle phase difference matrix and a multi-phase difference composite signal model in sequence for the IQ data frame model after time filtering processing; Step 5. Constructing a signal covariance matrix by using the multi-phase difference composite signal model, and applying spatial smoothing filtering to the signal covariance matrix; The step 5 is specifically: Applying spatial smoothing filter to the polyphase difference composite signal model, the filter result is represented by and the dimension of signal covariance matrix is reduced; To perform eigenvalue decomposition to obtain a reduced dimension signal subspace and a reduced dimension noise subspace, which is expressed in the formula as follows: ； wherein, denotes the reduced dimension signal subspace, consisting of the eigenvectors corresponding to the largest eigenvalues; denotes the reduced dimension noise subspace, consisting of the eigenvectors corresponding to the remaining small eigenvalues; is composed of the largest eigenvalues, is composed of the remaining small eigenvalues; The reduced-dimension signal steering vector is expressed by using the following formula: ； ； wherein denotes a signal steering vector in the azimuth direction, denotes a signal steering vector in the azimuth direction, and denotes an azimuth angle, denotes an elevation angle, denotes an antenna spacing, , denotes an overlapping subarray size, denotes a signal wavelength; j is the imaginary unit in the complex domain; The formula of the signal covariance matrix is: ； wherein denotes the mathematical expectation, denotes the conjugate transpose of the polyphase difference composite signal model; Step 6. Fusing the reduced-dimension signal steering vector obtained in step 5 and a reduced-dimension noise subspace into angle estimation.

2. The high-precision positioning method based on angle information fusion estimation according to claim 1, characterized in that, The step 1 is specifically: Suppose there are root antennas on a two-dimensional rectangular antenna array, where represents the number of antenna columns, represents the number of antenna rows, and suppose there are sources, set the antenna at the upper right corner as the reference antenna, number the reference antenna as , number the rest of the antennas in the order of , and number the source as . Establish a Cartesian right-hand coordinate system with the reference antenna as the origin to construct a reference antenna coordinate model.

3. The high-precision positioning method based on angle information fusion estimation according to claim 2, characterized in that, The IQ data frame model of step 2 divides data into reference IQ data and angle IQ data; the reference IQ data is all at reference antennas The up-sampling is expressed by the following formula: ； wherein, the reference data matrix, is collected during a sampling period, the I and Q values, are collected during a reference sampling period, denotes the number of reference IQ data set samples; The angle IQ data is expressed by using the following formula: ； in, Refers to the antenna snapshot period The angle data matrix below, and the antenna snapshot period refer to the time taken to sequentially collect data from all antennas once. Indicates the number of antenna snapshots. Indicates the angle sampling period The I and Q values ​​are as follows. Indicates the number is antenna, Indicates which sampling protocol to select; At the sampling period The complete IQ data frame model is: 。 4. The high-precision positioning method based on angle information fusion estimation according to claim 3, characterized in that, The step 3 is specifically: The time filtering is applied to the reference IQ data and the angle IQ data respectively, and the filtered IQ data frame model is expressed by using the following formula: ； wherein, represents the complete IQ data frame model after applying time filtering, represents the sampling period, represents the reference IQ data after applying time filtering, whose formula is: ； For the angle IQ data after time filtering application, the formula is expressed as follows: 。 5. The high-precision positioning method based on angle information fusion estimation according to claim 4, characterized in that, The step 4 is specifically: The phase solving calculation formula is applied to the IQ data frame model after time filtering processing, so as to obtain a phase matrix; The phase solving calculation formula is: ； wherein denotes the inverse tangent function; The phase matrix is: ； wherein is calculated from a reference phase matrix, is calculated from an angular phase matrix; According to the reference phase matrix, the reference phase difference calculation formula is used for calculation: ； In the formula, p represents a serial number of the reference phase difference data; Applying the data normalization calculation formula and the reference phase difference calculation formula to the reference phase matrix, the reference phase difference data is restricted to... The calculated reference phase difference matrix is ​​shown below: ； The mean reference phase difference is calculated by using a mean reference phase difference calculation formula as follows: ； The angle phase is differentially calculated with a reference antenna angle phase, and the mean reference phase difference is applied in combination, so as to obtain a calculation formula of the mean reference phase difference and the angle phase difference in combination, and then a data normalization calculation formula is applied to the calculation formula; the mean reference phase difference which can truly reflect the reference antenna is used as a first element of the angle phase difference matrix, and a calculation formula of the angle phase difference matrix in combination is as follows: ； wherein is the angular phase difference matrix, is the angular phase of an arbitrary antenna, is the angular phase of the reference antenna, is the mean reference phase difference, is the sampling switching time, denotes the set of antenna sequence numbers; Δφ * is the phase difference normalization processing operator; The angle phase difference matrix is obtained by calculation: ； A multi-phase difference composite signal model is constructed and expressed as follows: 。 6. The high-precision positioning method based on angle information fusion estimation according to claim 1, characterized in that, The step 6 is specifically: The reduced-dimension signal steering vector obtained in step 5 and the reduced-dimension noise subspace are fused into angle estimation, and the formula is expressed as follows: ； wherein denotes a spatial spectrum function, denotes a Kronecker product; The spatial spectrum function is normalized and logarithmized, and the formula is expressed as follows: ； wherein, represents a normalized and logarithmized spatial spectrum function, represents the maximum spectral peak in the spatial spectrum function; the source angle is estimated by finding the azimuth and elevation angles corresponding to the maximum spectral peak, i.e. the source angle estimation value.

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