Joint estimation method of azimuth and elevation angles of uniform array based on information theory

Abstract

The present invention discloses a method for joint estimation of azimuth and elevation angles of a uniform array based on information theory, which belongs to the field of array signal processing. The method comprises the following steps: constructing a multi-dimensional conditional probability density function of a uniform array using the properties of noise samples in array received signals; deriving a joint probability density function of the array received signal and the azimuth and elevation angles of a source using an information theory method; deriving a joint posterior probability density function of the azimuth and elevation angles of the uniform array based on the Bayesian formula; and performing a spectrum peak search on the constructed posterior probability density function to simultaneously obtain the azimuth and elevation angles of the target source. The present invention derives a posterior probability density function for joint estimation of azimuth and elevation angles of a uniform array based on information theory, and organically combines all available information using the Bayesian theory, thereby providing a more comprehensive and integrated joint angle estimation, and particularly improving the accuracy of joint angle estimation in a low signal-to-noise ratio range.

Description

Technical Field

[0001] This invention relates to the field of array signal processing technology, and in particular to a method for joint estimation of azimuth and elevation angles of a uniform area array based on information theory. Background Technology

[0002] The application of joint estimation of azimuth and elevation angles of sensor arrays covers a wide range of key fields. The application of this technology is mainly concentrated in radar systems, sonar systems, communication systems, aerospace, unmanned systems and earth observation.

[0003] Most existing methods for joint estimation of azimuth and elevation angles for uniform area arrays utilize high-resolution direction-finding algorithms such as MUSIC, RD-MUSIC, ESPRIT, and PM. These algorithms provide accurate signal estimations under ideal conditions, and have unique advantages, especially for signals with small angle separation. However, under low signal-to-noise ratio conditions, these algorithms typically exhibit limited robustness, are sensitive to noise, and are easily affected by significant factors. Summary of the Invention

[0004] This invention provides a method for joint estimation of azimuth and elevation angles of uniform arrays based on information theory, offering a more accurate estimation method for two-dimensional joint angle estimation of uniform arrays at low signal-to-noise ratio (SNR).

[0005] This invention provides a method for joint estimation of azimuth and elevation angles of a uniform array based on information theory, comprising the following steps:

[0006] Step 1) Construct the multidimensional conditional probability density function of the uniform array using the properties of noise samples in the array received signal of the uniform array, and derive the joint probability density function of the array received signal with the source azimuth angle and the source elevation angle using information theory methods.

[0007] Step 2): Simplify the joint probability density function of the uniform array according to the Bessel function, and derive the joint posterior probability density function of the source azimuth and source elevation angle of the uniform array under a given received signal by combining the Bayesian formula with the joint probability density function.

[0008] Step 3) Perform a local maximum search on the joint posterior probability density function to obtain the estimated values ​​of the source azimuth and source elevation angles of the uniform array.

[0009] Optionally, in one embodiment of the present invention, in step 1), the uniform array has a total of M×N uniformly distributed array elements, where N and M are the number of array elements on the x-axis and y-axis, respectively. The distance between adjacent array elements on the x-axis and y-axis is d, and d≤λ / 2, where λ is the wavelength. Each column of array elements is called a subarray, so there are subarrays 1, 2, ..., n, ..., M, for a total of M subarrays.

[0010] If a single far-field narrowband constant-mode scattered signal is incident on the uniform array with a two-dimensional wave direction of arrival (θ, φ) in a certain space, then the array-received signal X(t) of the uniform array is expressed as:

[0011] X(t)=A(θ,φ)s+N(t)

[0012] in, This represents a constant-mode scattering signal, where α is the amplitude of the scattering coefficient. θ is the scattering phase; N(t) is the additive complex white Gaussian noise with a mean of 0 and a variance of N0 on the uniform array; A(θ,φ) is the direction vector of the uniform array, where θ is the source elevation angle and φ is the source azimuth angle.

[0013] Since the noise N(t) is an additive complex Gaussian white noise sample, the source elevation angle θ, source azimuth angle φ, and scattering phase... Given the given conditions, the multidimensional conditional probability density function of the array-received signal x is expressed as:

[0014]

[0015] Based on information theory formulas, the joint probability density function of the received signal x of the uniform area array, the source azimuth angle φ, and the source elevation angle θ is derived as follows:

[0016]

[0017] in, Constant mode scattering phase The probability of constant mode scattering phase It follows a uniform distribution within the observation interval.

[0018] Optionally, in one embodiment of the present invention, in step 2), simplifying the joint probability density function of the uniform array according to the Bessel function includes:

[0019] Constant mode scattering phase Follows a uniform distribution, probability It is a constant, according to the definition of Bessel function and The joint probability density function p(x|φ,θ) of the array received signal x of the uniform area array, the source azimuth angle φ, and the elevation angle θ is expressed as:

[0020]

[0021] in, Represents a constant; I0(·) represents the zeroth-order Bessel function of the first kind; Re(·) and Im(·) represent taking the real and imaginary parts of the complex number, respectively; (·) H This indicates the conjugate transpose of the matrix; ||·||2 represents the 2-norm.

[0022] Optionally, in one embodiment of the present invention, in step 2), the joint posterior probability density function of the uniform array for a given received signal is derived based on Bayes' theorem and the joint probability density function, including:

[0023] Assuming that the source azimuth angle φ and source elevation angle θ follow a uniform distribution within the observation interval, the probabilities p(φ) and p(θ) of the source azimuth angle φ and source elevation angle θ are both constants. According to Bayes' theorem, the joint posterior probability density function of the source azimuth angle φ and source elevation angle θ for the uniform array under a given received signal x is:

[0024]

[0025] Optionally, in one embodiment of the present invention, in step 3),

[0026] Given the search range of the azimuth angle φ and elevation angle θ of the target signal source, a spectral peak search is performed on the joint posterior probability density function p(φ,θ|x) of the azimuth angle φ and elevation angle θ under a given received signal x. The azimuth angle and elevation angle corresponding to the local maximum value of the spectral function jointly estimated by the azimuth angle φ and elevation angle θ of the uniform array are obtained, which are the estimated azimuth angle and estimated elevation angle of the target signal source.

[0027] In the information-theory-based method for joint estimation of azimuth and elevation angles of a uniform array, this invention only requires the posterior probability density function of the estimation parameters to perform joint angle estimation of the azimuth and elevation angles of the uniform array. This provides a two-dimensional joint angle estimation method for uniform array systems that is independent of the algorithm. Simultaneously, the posterior probability density function considers all available information, including observation data, prior information, and the relationship between the estimation parameters. This information is organically combined through Bayesian theory, thus providing a more comprehensive and integrated estimate. By incorporating noise models and prior information, more accurate estimation results can be provided, especially under low signal-to-noise ratio conditions. Using the posterior probability density function for joint estimation of azimuth and elevation angles in the array provides more comprehensive, accurate, and robust estimation results. Compared with traditional joint estimation algorithms, such as MUSIC, ESPRIT, and PM algorithms, it considers multiple aspects of information, provides information on the uncertainty of parameter estimation, and can better handle complex estimation problems.

[0028] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the following description of the embodiments taken in conjunction with the accompanying drawings, wherein:

[0030] Figure 1 This is a flowchart of a method for jointly estimating the azimuth and elevation angles of a uniform array based on information theory, according to an embodiment of the present invention.

[0031] Figure 2 This is a uniform array model diagram according to an embodiment of the present invention;

[0032] Figure 3 A scatter plot comparing the method of the present invention and the MUSIC algorithm for joint estimation of azimuth and elevation angles;

[0033] Figure 4 The method of the present invention is shown in the spectral peak diagrams under different SNR conditions;

[0034] Figure 5 This is a top view of the spectral peak diagrams of the method in the embodiment of the invention under different SNR conditions;

[0035] Figure 6 The RMSE curves of the method for joint estimation of azimuth and elevation angles using the RD-MUSIC, ESPRIT, and 2D-PM algorithms in the embodiments of the invention are shown as a function of SNR. Detailed Implementation

[0036] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0037] The following describes an embodiment of the information theory-based joint estimation method for azimuth and elevation angles of a uniform array, with reference to the accompanying drawings. Addressing the issue mentioned in the background section that joint estimation algorithms for azimuth and elevation angles of uniform arrays typically exhibit limited robustness under low signal-to-noise ratio (SNR) conditions, are sensitive to noise, and are easily affected by significant noise, this invention proposes an information theory-based joint estimation method for azimuth and elevation angles of a uniform array, leveraging the unified framework and applicability to various signal and noise types inherent in information theory. Simulation comparisons verify that the proposed method outperforms other joint angle estimation algorithms under low SNR conditions. Furthermore, the method, based on modeling the statistical characteristics of the signal, exhibits better robustness and can provide relatively accurate estimates even under high noise levels.

[0038] Specifically, Figure 1 This is a flowchart illustrating a method for jointly estimating the azimuth and elevation angles of a uniform array based on information theory, according to an embodiment of the present invention.

[0039] like Figure 1 As shown, the information-theoretic-based method for joint estimation of azimuth and elevation angles of a uniform array includes the following steps:

[0040] Step 1) Construct a multidimensional conditional probability density function for the uniform array by utilizing the properties of noise samples in the array received signal. Then, derive the joint probability density function of the array received signal, source azimuth angle, and source elevation angle using information theory methods.

[0041] Specifically, a multidimensional conditional probability density function is constructed using the array-received signal X(t) from a uniform array. The joint probability density function p(x|φ,θ) of the array received signal x, the source azimuth angle φ, and the elevation angle θ is derived using information theory.

[0042] like Figure 2 As shown, the sensor array is a uniform surface array with a total of M×N uniformly distributed array elements. N and M are the number of array elements on the x-axis and y-axis, respectively. The distance between adjacent array elements on the x-axis and y-axis is d, and d≤λ / 2, where λ is the wavelength. In addition, the array element set in each column is called a subarray, so there are subarrays 1, 2, ..., n, ..., M, for a total of M subarrays.

[0043] Suppose there are K far-field narrowband constant-mode scattered signals in a certain space, arriving at a two-dimensional direction (θ). k ,φ k ) is incident on a uniform array, where (θ k ,φ k Let θ and y represent the elevation and azimuth angles of the k-th (k = 1, 2, ..., K) signal source, respectively. Then, the array received signal X(t) of the uniform area array can be expressed as:

[0044]

[0045] in: Represents the Kronecker product; Let A be the direction vector of a uniform area array, denoted as the direction matrix A. x and A y Khatri-Rao product;

[0046] Where the direction matrix A x and A y They are represented as follows:

[0047]

[0048]

[0049] in:

[0050] The signal source direction vector on the x-axis;

[0051] The signal source direction vector is on the y-axis;

[0052] Represents the constant-mode scattering signal source vector; This represents a constant-mode scattering signal, where α is the amplitude of the scattering coefficient, typically a constant. The scattering phase;

[0053] It is an additive complex white Gaussian noise with a mean of 0 and a variance of N0 on a uniform surface array.

[0054] For ease of study, this method only considers the single-source case. Also, for simplicity, X(t) is denoted as:

[0055] X = As + N

[0056] Since the noise N(t) is an additive complex Gaussian white noise sample, at elevation angle θ, azimuth angle φ, and scattering phase... Given the given conditions, the multidimensional conditional probability density function of the array-received signal x is expressed as:

[0057]

[0058] Therefore, based on information theory formulas, the joint probability density function of the received signal x of the uniform area array, the source azimuth angle φ, and the elevation angle θ can be derived as follows:

[0059]

[0060] in, Constant mode scattering phase The probability of constant mode scattering phase It follows a uniform distribution within the observation interval.

[0061] Step 2) Simplify the joint probability density function of the uniform array using Bessel functions. Based on Bayes' theorem and the joint probability density function, derive the joint posterior probability density function of the source azimuth and elevation angles of the uniform array under a given received signal. This is the spectral function of the joint estimation of the azimuth and elevation angles of the uniform array.

[0062] First, prior assumptions are made regarding the constant-mode scattering information, the azimuth angle of the target source, and the elevation angle. The constant-mode scattering phase is assumed. If is a random variable that follows a uniform distribution within the observation interval [0, 2π], then the scattering phase... The prior probability density function is expressed as:

[0063]

[0064] Assuming the elevation angle θ of the target source follows a uniform distribution within the observation interval [-|Θ| / 2, |Θ| / 2|], then the prior probability density function of the elevation angle θ is:

[0065]

[0066] Assuming the azimuth angle φ of the target source follows a uniform distribution within the observation interval [-|Φ| / 2,|Φ| / 2], then the prior probability density function of the elevation angle θ is:

[0067]

[0068] Where |·| represents the logarithmic modulo value.

[0069] According to the definition of Bessel functions and The joint probability density function p(x|φ,θ) of the received signal x from the uniform area array, along with the source azimuth angle φ and elevation angle θ, can be further expressed as:

[0070]

[0071] in, Represents a constant; I0(·) represents the zeroth-order Bessel function of the first kind; Re(·) and Im(·) represent taking the real and imaginary parts of the complex number, respectively; (·) H This indicates the conjugate transpose of the matrix; ||·||2 represents the 2-norm.

[0072] Assuming that the azimuth angle φ and elevation angle θ of the target information source follow a uniform distribution within the observation interval, then their probabilities p(φ) and p(θ) are both constants.

[0073] According to Bayes' theorem, the joint posterior probability density function of the azimuth angle φ and elevation angle θ of a uniform array under a given received signal x can be obtained as follows:

[0074]

[0075] Step 3) Perform a local maximum search on the joint posterior probability density function to obtain the estimated values ​​of the source azimuth and source elevation angles of the uniform array.

[0076] Based on the spectral function p(φ,θ|x) jointly estimated by the azimuth and elevation angles of the uniform array, the spectral peak search algorithm is used to obtain the estimated values ​​of the azimuth and elevation angles of the uniform array.

[0077] Furthermore, given the search range of the azimuth angle φ and elevation angle θ of the target signal source, a spectral peak search is performed on the joint posterior probability density function p(φ,θ|x) of the azimuth angle φ and elevation angle θ under a given received signal x. This yields the azimuth angle and elevation angle corresponding to the local maximum value of the spectral function jointly estimated by the azimuth angle φ and elevation angle θ of the uniform array, which are the estimated azimuth angle and estimated elevation angle of the target signal source.

[0078] For a given snapshot, assuming the two-dimensional direction of arrival of the target source is (φ0, θ0), the received signal of the uniform array... Substituting these values ​​into the spectral function p(φ,θ|x) for the joint estimation of azimuth and elevation angles, we can obtain:

[0079]

[0080] Where: ρ 2 =α 2 / N0 represents the SNR of each element in a uniform array. G(φ,θ)=A H (φ,θ)A(φ0,θ0) represents the array pattern of a uniform surface array, F(φ,θ,x)=A H (φ,θ)x represents the cross-correlation function between the signal and noise, indicating the influence of noise. Therefore, G(φ,θ) and F(φ,θ,x) together determine the characteristics of the spectral function p(φ,θ|x) for the joint estimation of azimuth and elevation angles.

[0081] Given the search range of the azimuth angle φ and elevation angle θ of the target source, the top K largest spectral peaks in the search area can be obtained by using the grid exhaustive search method. The corresponding azimuth angle and elevation angle are the estimated values ​​of the azimuth angle φ and elevation angle θ of the target source, where K represents the number of target sources.

[0082] Figure 3 This is a scatter plot comparing the method of the invention embodiment with the MUSIC algorithm for joint estimation of azimuth and elevation angles. The simulation conditions are: number of snapshots J = 50, number of sources K = 1, number of simulations T = 50, number of array elements in the uniform array N = 16, M = 16, and the actual positions of the sources φ0 = 45°, θ0 = 45°. Figure 3 As can be observed, the information-theory method, by modeling the statistical properties of the signal, performs well in complex signal scenarios and low signal-to-noise ratio (SNR) environments. In the scatter plot, the data points of the information-theory method are relatively evenly distributed, and the error compared to the true values ​​is small. This indicates that, under given experimental conditions, the information-theory method has high accuracy and robustness in the joint estimation of azimuth and elevation angles. The MUSIC algorithm is a high-resolution spectrum estimation method suitable for signals separated by small angles. The figure shows that the MUSIC algorithm's data points demonstrate high resolution for small-angle targets, but it performs better at high SNR. In low SNR conditions, the data point distribution may be more scattered, indicating weaker robustness in complex environments. A comprehensive comparison of the two methods reveals that the information-theory method can compensate for the disadvantages of high-resolution spectrum estimation methods in low SNR environments, exhibiting better robustness.

[0083] Figure 4 This is a spectral peak diagram of the method in an embodiment of the invention under different SNR conditions. The simulation conditions are: number of snapshots J = 1, number of sources K = 1, number of array elements in the uniform area array N = 16, M = 16, and actual positions of the sources φ0 = 45°, θ0 = 45°. From Figure 4 As can be seen, at signal-to-noise ratios (SNRs) of 10 dB, 5 dB, -5 dB, and -10 dB, the spectral peaks of the method for joint estimation of azimuth and elevation angles of a uniform array exhibit peaks of varying intensities and distributions. Under high SNR conditions, the peaks are sharper and more intense, reflecting the accuracy and confidence of the estimation. As the SNR decreases, the peaks gradually become blurred and their intensity weakens, indicating that under low SNR conditions, the estimation results are significantly affected by noise. However, the information-theoretic method can still accurately estimate the target source location, and a clear peak is observed, reflecting the applicability of the information-theoretic method for joint estimation of azimuth and elevation angles.

[0084] Figure 5This is a top view of the spectral peak diagram of the method under different SNR conditions according to an embodiment of the invention. The simulation conditions are: number of snapshots J=1, number of sources K=1, number of array elements in the uniform area array N=16, M=16, and actual positions of the sources φ0=45°, θ0=45°. Figure 5 To more comprehensively evaluate the estimation performance, a top-down view of the spectral peak plot was further analyzed. Ground truth points were added to the top view of the spectral peak plot for direct comparison with the estimated peak values. These ground truth points represent the actual azimuth and elevation angles of the target source. By comparing the ground truth points with the estimated peak values, the consistency between the estimation results and the actual situation can be more clearly understood, demonstrating the estimation accuracy of the information-theoretic method for jointly estimating the azimuth and elevation angles of a uniform array.

[0085] Figure 6 The RMSE curves of the performance of the method of this invention in joint estimation of azimuth and elevation angles with SNR, compared with RD-MUSIC, ESPRIT, and 2D-PM algorithms, are shown. Simulation conditions were: number of snapshots J = 100, number of sources K = 1, number of simulations T = 100, number of elements in the uniform array N = 8, M = 8, and actual source positions φ0 = 45°, θ0 = 60°. Figure 6 As can be observed, under different signal-to-noise ratio (SNR) conditions, the information-theory method exhibits a lower RMSE value across the entire SNR range. This indicates that the method has high accuracy in estimating azimuth and elevation angles, especially maintaining relatively low errors in low SNR environments. While the RD-MUSIC, ESPRIT, and 2D-PM algorithms perform better under high SNR conditions, their RMSE increases significantly at low SNRs. This suggests that the RD-MUSIC, ESPRIT, and 2D-PM algorithms are more suitable for high SNR environments, but their performance degrades under high noise conditions. By comparing the RMSE curves, it can be concluded that the information-theory method has more robust performance than other algorithms under different SNR conditions. In low SNR environments, the information-theory method significantly outperforms the RD-MUSIC, ESPRIT, and 2D-PM algorithms. This demonstrates the adaptability of the information-theory method to noisy environments in practical applications, providing a more reliable estimation method for the joint estimation of azimuth and elevation angles of uniform arrays.

[0086] This invention proposes an information-theoretic-based method for joint estimation of azimuth and elevation angles of a uniform array. The information-theoretic approach, based on a probabilistic framework, can handle uncertainties more flexibly. By modeling the probability density function, it can provide a confidence measure of the estimate, especially in low signal-to-noise ratio (SNR) environments, better handling of uncertainties and improving the robustness of the estimation. Furthermore, the information-theoretic method demonstrates advantages over traditional joint angle estimation algorithms in terms of global optimization, probabilistic framework, multi-dimensional information synthesis, adaptability, and performance under low SNR conditions. Simulation comparisons with the RMSE of joint angle estimation using traditional algorithms verify the rationality and accuracy of the proposed method.

[0087] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., refer to specific features, structures, materials, or characteristics described in connection with that embodiment or example, which are included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0088] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this invention, "N" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0089] Any process or method description in the flowchart or otherwise herein can be understood as representing a module, segment, or portion of code comprising one or N executable instructions for implementing custom logic functions or processes, and the scope of preferred embodiments of the invention includes additional implementations in which functions may be performed not in the order shown or discussed, including substantially simultaneously or in reverse order depending on the functions involved, as will be understood by those skilled in the art to which embodiments of the invention pertain.

Claims

1. A method for jointly estimating the azimuth and elevation angles of a uniform array based on information theory, characterized in that: The following steps are involved: Step 1), constructing a multidimensional conditional probability density function of the uniform array using the properties of noise samples in the array received signal of the uniform array, and deriving a joint probability density function of the array received signal and the source azimuth angle and the source elevation angle by using an information theory method; Step 2), simplifying the joint probability density function of the uniform array according to the Bessel function, and deriving the joint posterior probability density function of the source azimuth angle and the source elevation angle of the uniform array under a given received signal according to the Bayesian formula combined with the joint probability density function; Step 3), performing a local maximum search on the joint posterior probability density function to obtain estimated values ​​of the source azimuth angle and the source elevation angle of the uniform array; In step 1), the uniform array has a total of M×N uniformly distributed array elements, N and M are the number of array elements on the x-axis and the y-axis respectively, the spacing between adjacent array elements on the x-axis and the y-axis is d, and d≤λ / 2, λ is the wavelength, and the set of array elements in each column is called a subarray, then there are subarrays 1, 2, ..., n, ..., M, and a total of M subarrays; Assuming that a single far-field narrowband constant mode scattered signal is incident on the uniform array in a two-dimensional wave direction (θ, φ) in a certain space, the array receiving signal X(t) of the uniform array is expressed as: X(t)=A(θ,φ)s+N(t) in, represents the constant mode scattering signal, α is the amplitude of the scattering coefficient, is the scattering phase; N(t) is the additive complex Gaussian white noise with a mean of 0 and a variance of N0 on the uniform array; A(θ,φ) is the direction vector of the uniform array, θ is the source elevation angle, and φ is the source azimuth angle; Since the noise N(t) is an additive complex Gaussian white noise sample, the source elevation angle θ, source azimuth angle φ and scattering phase Under known conditions, the multidimensional conditional probability density function of the array receiving signal x is expressed as: Based on the information theory formula, the joint probability density function of the uniform array receiving signal x, the source azimuth angle φ and the source elevation angle θ is derived as follows: in, is the constant mode scattering phase The probability of constant mode scattering phase It obeys a uniform distribution within the observation interval.

2. The method according to claim 1, characterized in that In step 2), the joint probability density function of the uniform array is simplified according to the Bessel function, including: Constant Mode Scattering Phase Follows uniform distribution, probability is a constant, according to the definition of Bessel function and The joint probability density function p(x|φ,θ) of the array receiving signal x of the uniform array and the source azimuth angle φ and elevation angle θ is expressed as: in, represents a constant; I0(·) represents the first-kind zero-order Bessel function; Re(·) and Im(·) represent the real and imaginary parts of a complex number, respectively; (·) H represents the conjugate transpose of the matrix; ||·||2 represents the 2-norm.

3. The method according to claim 2, characterized in that In step 2), the joint posterior probability density function of the source azimuth angle and the source elevation angle of the uniform array under a given received signal is derived according to the Bayesian formula combined with the joint probability density function, including: Assuming that the source azimuth angle φ and the source elevation angle θ obey uniform distribution in the observation interval, the probabilities p(φ) and p(θ) of the source azimuth angle φ and the source elevation angle θ are both constants. According to the Bayesian formula, the joint posterior probability density function of the source azimuth angle φ and the source elevation angle θ of the uniform array under the given received signal x is obtained as follows:

4. The method according to claim 3, characterized in that In step 3), under the condition that the search range of the azimuth angle φ and the elevation angle θ of the given target signal source is given, a spectral peak search is performed on the joint posterior probability density function p(φ,θ|x) of the azimuth angle φ and the elevation angle θ under the given received signal x, and the azimuth angle and the elevation angle corresponding to the local maximum value of the spectral function of the joint estimation of the azimuth angle φ and the elevation angle θ of the uniform array are obtained, which are the estimated azimuth angle and the estimated elevation angle of the target signal source.

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