Information theory based non-circular sparse array DOA estimation performance evaluation method

Abstract

The application discloses a non-circular sparse array DOA estimation performance evaluation method based on information theory, which comprises the following steps: firstly, a multi-dimensional probability density function (PDF) of a received signal is constructed, a joint PDF of the received signal and DOA and a DOA posteriori PDF are derived through information theory, and a system DOA information quantity is obtained by simplifying the DOA posteriori PDF with a Bessel function; then, a DOA posteriori PDF of given noise is derived, and a DOA information approximate upper limit is obtained by simplifying the DOA posteriori PDF with a Taylor expansion; finally, a DOA estimation performance index entropy error is obtained by using the posteriori differential entropy; the application builds a non-circular sparse array system DOA information theory framework based on information theory, in actual signal processing, the entropy error of a parameter can be calculated only by estimating the posteriori PDF of the parameter, and a performance limit independent of an algorithm is provided; in addition, it is found through simulation that the DOA information quantity approaches the DOA information upper limit under a high signal-to-noise ratio, and the entropy error approaches the Cramer-Rao limit, thereby verifying the rationality of the index.

Description

Technical Field

[0001] This invention relates to the field of DOA estimation technology in array signal processing, and in particular to a method for evaluating the performance of DOA estimation for non-circular sparse arrays based on information theory. Background Technology

[0002] The core of array signal processing is to arrange a specified number of arrays in a certain way at fixed positions in a space, and extract useful feature information from the source data received by multiple array elements, resulting in higher spatial resolution. Compared with single sensors, array signal processing has advantages in anti-interference capability and signal gain, and is attracting increasing attention.

[0003] Non-circular signals are common in modern communication systems. A non-circular signal is defined as one whose elliptic covariance matrix is ​​non-zero. This characteristic allows for the virtual doubling of the number of array elements, enabling the resolution of more signal sources. Furthermore, sparse arrays offer a larger array aperture and more degrees of freedom for the same number of elements, improving direction-finding accuracy, resolution, and the maximum number of signal sources that can be processed. They also require fewer elements for the same array aperture, significantly reducing equipment costs. Additionally, they reduce mutual coupling between elements, mitigating their impact on direction-finding performance.

[0004] Shannon theory, as the theoretical foundation of communication technology, reveals the essence of information and the basic laws of transmission. Information theory introduces the concept of "entropy" to quantify the amount of information, while mutual information represents the change in entropy caused by the presence of other random variables. However, research on array signal processing using Shannon theory mainly focuses on estimating the number of information sources, with only preliminary theoretical analysis of the information acquisition process for array parameter estimation, especially for non-circular sparse array systems, where a unified theoretical framework has not yet been formed. Therefore, to address these issues, this invention, based on information theory, derives the relationship between the change in information content during parameter estimation and the performance of the array system, and proposes an information theory-based method for evaluating the performance of direction of arrival (DOA) estimation for non-circular sparse arrays. Summary of the Invention

[0005] To address the above technical problems, this invention provides a method for evaluating the performance of non-circular sparse matrix DOA estimation based on information theory, comprising the following steps:

[0006] S1. Construct a multidimensional probability density function using the received signal x(t) through a sparse array. The joint probability density function p(x,θ) of the sparse array received signal x(t) and the source azimuth angle θ is derived using information theory methods.

[0007] S2. Derive the DOA posterior probability density function p(θ|x) of the noncircular sparse matrix based on the joint probability density function p(x,θ), and simplify it using Bessel functions;

[0008] S3. Obtain the DOA information I(X; Θ) of the non-circular sparse array according to the mutual information formula;

[0009] S4. Using the joint probability density function p(x,θ) of DOA, derive the posterior probability density function p(θ|n) of DOA under given noise conditions. Simplify using Taylor expansion, and obtain the information-theoretic approximate upper bound I of the non-circular sparse array DOA according to the differential entropy formula for variables following a Gaussian distribution. aub (X;Θ);

[0010] S5. Using the posterior differential entropy h(Θ|X), we obtain an index for evaluating the DOA estimation performance of a non-circular sparse array system, namely, the entropy error. And derive the CRB of non-circular nested arrays, i.e. Will and By comparison, the performance evaluation of the non-circular sparse array system in the entire signal-to-noise ratio range is analyzed.

[0011] The technical solution further defined in this invention is:

[0012] Furthermore, in step S1, the sparse array is a general nested array with a total number of array elements of N. The general nested array includes a dense subarray and a sparse subarray. The dense subarray has N1 array elements and the array element spacing is d0. The sparse subarray has N2 array elements and the array element spacing is (N1+1)d0. Where d0=λ / 2, λ is the wavelength, and the total number of array elements N=N1+N2.

[0013] The aforementioned information-theoretic-based method for evaluating the performance of DOA estimation for non-circular sparse arrays, in step S1, assumes that a single far-field narrowband non-circular signal is incident on a general nested array with a direction of arrival θ in a certain space, and defines the received signal x(t) as:

[0014] x(t)=a(θ)s(t)+n(t)

[0015] The non-circular signal is s(t) = Φs R (t), s R (t) represents a real-valued signal. Contains non-circular information, where j represents an imaginary number. θ represents the non-circular phase; n(t) is additive white Gaussian noise with zero mean and variance N0 on a general nested matrix; a(θ) is the direction vector of the general nested matrix;

[0016] Since the noise n(t) is additive white Gaussian noise, at the source azimuth θ and in non-circular phase... Given the given conditions, the multidimensional probability density function of the array-received signal x is:

[0017]

[0018] in, The non-circular phase is represented by N, the total number of array elements is N, a(θ) is the direction vector of a general nested array, N0 is the variance of additive white Gaussian noise, and s and s R Let (·) represent a non-circular signal and a real-valued signal, respectively. H Indicates conjugate transpose;

[0019] The joint probability density function of the array received signal x and the source azimuth θ is derived.

[0020]

[0021] in, Indicates a known non-circular phase Under the condition of , the probability density function of the array receiving signal x; Indicates non-circular phase The probability density function.

[0022] The aforementioned information-theoretic-based method for evaluating the performance of non-circular sparse matrix DOA estimation, in step S2, involves the non-circular sparse matrix DOA information θ and phase... All follow a uniform distribution, so their probabilities are all constants; based on p(x,θ), the posterior probability density function of the DOA of the non-circular sparse matrix is ​​obtained as follows:

[0023]

[0024] Where ||Θ|| represents the length of the DOA observation interval; (·) H Denotes the conjugate transpose; x in the multidimensional probability density function of the array received signal x H The value of x depends on the actual value of the source DOA, so it is omitted in the above formula.

[0025] The aforementioned information-theoretic-based method for evaluating the performance of DOA estimation for non-circular sparse matrices, in step S2, simplifies the numerator in the expression for the posterior probability density function of the DOA of the non-circular sparse matrix to obtain:

[0026]

[0027] Where Im(·) represents the imaginary part of the complex number; substituting the above equation into the expression for the posterior probability density function of the DOA of a non-circular sparse matrix, we get:

[0028]

[0029] Where I0(·) represents the Bessel function.

[0030] In the aforementioned information-theoretic-based method for evaluating the performance of non-circular sparse matrix DOA estimation, step S3 involves obtaining the DOA information content as the difference between the differential entropy of the prior and posterior distributions of the DOA, based on the formula for mutual information.

[0031]

[0032] Among them, E x [·] indicates the desired outcome.

[0033] In the aforementioned information-theoretic-based non-circular sparse matrix DOA estimation performance evaluation method, step S4 derives the DOA posterior probability density function p(θ|x) under given noise conditions using the DOA posterior probability density function formula p(x,θ):

[0034]

[0035] Where I0(·) represents the Bessel function, (·) H This indicates taking the conjugate transpose, where θ0 is the assumed actual direction of the signal received by the array. Given non-circular phase information;

[0036] Furthermore, the approximate upper bound of the DOA information content of non-circular nested matrices is obtained as follows:

[0037]

[0038] Where, σ 2 Let p(θ|n) be the variance of the posterior probability density function of DOA under given noise conditions, which follows a Gaussian distribution.

[0039] In the aforementioned information-theoretic-based method for evaluating the performance of non-circular sparse matrix DOA estimation, step S5 uses the posterior differential entropy h(Θ|X) to represent the uncertainty of the unknown parameter Θ given the vector X, which is used to evaluate the estimation performance of the system. Therefore, the entropy error of the nested matrix is ​​used as an indicator to accurately evaluate the estimation performance of the nested array system, and its specific expression is as follows:

[0040]

[0041] in, This represents the entropy error.

[0042] In the aforementioned information-theoretic-based method for evaluating the performance of non-circular sparse matrix DOA estimation, step S5, the known CRB of the unbiased estimator θ is:

[0043]

[0044] in:

[0045]

[0046]

[0047] Based on the direction matrix of the nested matrix, the following expression is obtained:

[0048] a H (θ)a(θ)=N

[0049]

[0050]

[0051] Therefore, the DOA estimate CRB of the non-circular sparse matrix is ​​obtained as follows:

[0052]

[0053] in, Represents a non-circular nested array of CRBs.

[0054] The beneficial effects of this invention are:

[0055] In this invention, the entropy error of the estimated parameter can be calculated simply by providing the posterior probability density function, thus providing an algorithm-independent performance boundary for non-circular sparse array systems. Simultaneously, a closed-form expression for the DOA information content of non-circular nested arrays is derived based on the posterior probability density function, further obtaining an approximate upper bound for the system's DOA information under high signal-to-noise ratio conditions, and verifying the asymptotic nature of the upper bound through numerical simulation. Furthermore, for the case where the total number of array elements remains constant, the optimal array element configuration requirements for non-circular sparse array systems are obtained by changing the array element configuration in conjunction with the DOA information content. Compared with non-circular uniform linear array systems, the method of this invention further saves the hardware cost of array element construction and proposes a new performance evaluation index, entropy error, to assess the information acquisition capability of non-circular sparse array systems. Attached Figure Description

[0056] Figure 1 This is an overall flowchart of the present invention;

[0057] Figure 2 This is a schematic diagram of a typical nested array model in an embodiment of the present invention;

[0058] Figure 3 This is a schematic diagram illustrating the variation of DOA information content with SNR under different total number N of nested array elements according to an embodiment of the present invention;

[0059] Figure 4This is a schematic diagram illustrating the variation of DOA information with SNR under different array configurations in an embodiment of the present invention;

[0060] Figure 5 This is a schematic diagram comparing the amount of DOA information received by the array in an embodiment of the present invention with that in a non-circular uniform linear array system;

[0061] Figure 6 This is a schematic diagram illustrating the change of entropy error with SNR under different total number N of nested array elements in an embodiment of the present invention;

[0062] Figure 7 This is a schematic diagram comparing the entropy error of the method described in this invention with that of an array receiving information in a non-circular uniform linear array system. Detailed Implementation

[0063] This embodiment provides a performance evaluation method for DOA estimation of non-circular sparse matrices based on information theory, such as... Figure 1 As shown, it includes the following steps

[0064] S1. Construct a multidimensional probability density function using the received signal x(t) through a sparse array. The joint probability density function p(x,θ) of the sparse array received signal x(t) and the source azimuth angle θ is derived using information theory methods.

[0065] like Figure 2 As shown, the sparse array is a general nested array with a total number of N array elements. The general nested array includes a dense subarray and a sparse subarray. The dense subarray has N1 array elements and the array element spacing is d0. The sparse subarray has N2 array elements and the array element spacing is (N1+1)d0. Where d0=λ / 2, λ is the wavelength, and the total number of array elements N=N1+N2.

[0066] Suppose a single far-field narrowband non-circular signal is incident on a general nested array with a direction of arrival θ in a certain space. Define the received signal x(t) as:

[0067] x(t)=a(θ)s(t)+n(t)

[0068] The non-circular signal is s(t) = Φs R (t), s R (t) represents a real-valued signal. Contains non-circular information, where j represents an imaginary number. θ represents the non-circular phase; n(t) is additive white Gaussian noise with zero mean and variance N0 on a general nested matrix; a(θ) is the direction vector of the general nested matrix;

[0069] Since the noise n(t) is additive white Gaussian noise, at the source azimuth θ and in non-circular phase... Given the given conditions, the multidimensional probability density function (PDF) of the array-received signal x is:

[0070]

[0071] Where Re(·) denotes taking the real part of the complex number, The non-circular phase is represented by N, the total number of array elements is N, a(θ) is the direction vector of a general nested array, N0 is the variance of additive white Gaussian noise, and s and s R Let (·) represent a non-circular signal and a real-valued signal, respectively. H Indicates conjugate transpose;

[0072] In a given phase Under these conditions, averaging the phase yields:

[0073]

[0074] Where p(θ) represents the probability density function of the source azimuth θ;

[0075] Therefore, the joint probability density function of the array received signal x and the source azimuth θ is derived.

[0076]

[0077] in, Indicates a known non-circular phase Under the condition of , the probability density function of the array receiving signal x; Indicates non-circular phase The probability density function.

[0078] S2. Derive the DOA posterior probability density function p(θ|x) of the noncircular sparse matrix based on the joint probability density function p(x,θ), and simplify it using Bessel functions;

[0079] Non-circular sparse matrix DOA information θ and phase All follow a uniform distribution, so their probabilities are all constants; based on p(x,θ), the posterior probability density function of the DOA of the non-circular sparse matrix is ​​obtained as follows:

[0080]

[0081] Where ||Θ|| represents the length of the DOA observation interval; (·) H Denotes the conjugate transpose; x in the multidimensional probability density function of the array received signal x H The value of x depends on the actual value of the source DOA, so it is omitted in the above formula.

[0082] Simplifying the numerator in the expression for the posterior probability density function of a non-circular sparse matrix, we get:

[0083]

[0084] Where Im(·) represents the imaginary part of the complex number; substituting the above equation into the expression for the posterior probability density function of the DOA of a non-circular sparse matrix, we get:

[0085]

[0086] Where I0(·) represents the Bessel function.

[0087] S3. Obtain the DOA information I(X; Θ) of the non-circular sparse array according to the mutual information formula;

[0088] The amount of information received by the array can be obtained from the posterior probability density function of the DOA of a non-circular sparse array. Therefore, according to the formula for mutual information, the DOA information content is the difference between the differential entropy of the prior and posterior distributions of the DOA.

[0089]

[0090] Among them, E x [·] indicates the desired outcome.

[0091] S4. Using the joint probability density function p(x,θ) of DOA, derive the posterior probability density function p(θ|n) of DOA under given noise conditions. Simplify using Taylor expansion, and obtain the information-theoretic approximate upper bound I of the non-circular sparse array DOA according to the differential entropy formula for variables following a Gaussian distribution. aub (X;Θ).

[0092] Assuming the actual direction of the received signal in a typical nested array is θ0, and the non-circular phase parameter is... The following is a sample of the received signals from the actual array:

[0093]

[0094] By combining the sample of the actual received signal with the expression p(x,θ) of the DOA posterior probability density function of the non-circular sparse array, the DOA posterior probability density function p(θ|x) under given noise conditions is obtained as follows:

[0095]

[0096] in,(·) H This indicates taking the conjugate transpose; θ0 is the assumed actual direction of the signal received by the array. Given non-circular phase information; For the noise term, since the noise n is a complex random variable, the existence of the non-circular phase term will not affect the statistical properties of the noise and can be ignored. According to the above formula, it can be seen that the posterior probability density distribution p(θ|n) of DOA under given noise conditions mainly depends on the magnitude of the signal term and the noise term.

[0097] Since the signal term has a significant impact on the posterior probability density distribution p(θ|n) under high signal-to-noise ratio (SNR) conditions, the noise term can be ignored without affecting the characteristics of the posterior probability distribution. Therefore, the posterior probability density function p(θ|n) of the DOA under given noise conditions can be approximated as follows:

[0098]

[0099] in, The signal-to-noise ratio of the information source.

[0100] Since the direction vectors of the two subarrays of a typical nested matrix can be expressed as follows:

[0101]

[0102] The direction vector of the nested array is then represented as:

[0103]

[0104] Therefore, |a H The specific expression of (θ)a(θ0)| is as follows:

[0105]

[0106] Where ω=πd(sinθ-sinθ0) / λ, to facilitate obtaining an approximate result of the DOA information content, for |a H (θ)a(θ0)| is expanded using Taylor series at θ=θ0, resulting in

[0107] |a H (θ)a(θ0)|≈N-β 2 (θ-θ0) 2

[0108] in:

[0109]

[0110] Furthermore, the approximate expansion of the Bessel function I0(x) is:

[0111]

[0112] The approximate expression of the Bessel function and |aH Substituting the Taylor expansion approximation of (θ)a(θ0)| into the posterior probability density function p(θ|n) of DOA under given noise conditions, we can obtain:

[0113]

[0114] Where θ is a normalized constant coefficient, it can be concluded that p(θ|n) is approximately a Gaussian distribution, and its variance is expressed as σ. 2 =(4ρ 2 β 2 ) -1 .

[0115] Therefore, based on the differential entropy formula for variables following a Gaussian distribution, the approximate upper bound of the DOA information content of a non-circular nested matrix is ​​obtained as follows:

[0116]

[0117] Where, σ 2 Let p(θ|n) be the variance of the posterior probability density function of DOA under given noise conditions, which follows a Gaussian distribution.

[0118] S5. Using the posterior differential entropy h(Θ|X), we obtain an index for evaluating the DOA estimation performance of a non-circular sparse array system, namely, the entropy error. And the Cramér-Rao Bound (CRB) for non-circular nested arrays is derived, i.e. Will and By comparison, the performance evaluation of the non-circular sparse array system in the entire signal-to-noise ratio range is analyzed.

[0119] The posterior differential entropy h(Θ|X) represents the uncertainty of the unknown parameter Θ given the vector X, and is used to evaluate the estimation performance of the system. Therefore, the entropy error of the nested array is used as an indicator to accurately evaluate the estimation performance of the nested array system, and its specific expression is as follows:

[0120]

[0121] in, This represents the entropy error.

[0122] In estimation theory, the CRB is an important metric for evaluating the performance of an unbiased estimator. Given the CRB of the unbiased estimator θ:

[0123]

[0124] in:

[0125]

[0126]

[0127] Based on the direction matrix of the nested matrix, the following expression is obtained:

[0128] a H (θ)a(θ)=N

[0129]

[0130]

[0131] Therefore, the DOA estimate CRB for a non-circular sparse matrix can be obtained as follows:

[0132]

[0133] like Figure 3 The figure shows the variation of DOA information content with SNR under different total number of array elements N in the nested array according to the method of this embodiment. The simulation conditions are: number of snapshots J=1, number of information sources K=1, number of simulations T=1000, and the number of array elements in the dense subarray in the general nested array is N1=8, and the number of array elements in the sparse subarray is N2=[9,12,15].

[0134] like Figure 4 The figure shows the DOA information changing with SNR under different array configurations in this embodiment. The simulation conditions are: number of snapshots J=1, number of information sources K=1, number of simulations T=1000. Generally, the number of elements in the dense subarray in the nested array is N1=[9,7,5], and the number of elements in the sparse subarray is N2=[10,12,14].

[0135] like Figure 5 The diagram shows a comparison of the amount of DOA information received by the array in this embodiment with that in a non-circular uniform linear array system. The simulation conditions are: number of snapshots J=1, number of information sources K=1, number of simulations T=1000, the number of array elements in the dense subarray of a typical nested array is N1=8, the number of array elements in the sparse subarray is N2=9, and the number of array elements in the uniform linear array is M=17.

[0136] like Figure 6 The figure shows the change of entropy error with SNR under different total number of array elements N in the nested array method of this embodiment. The simulation conditions are: number of snapshots J=1, number of information sources K=1, number of simulations T=1000. Generally, the number of array elements in the dense subarray in the nested array is N1=8, and the number of array elements in the sparse subarray is N2=9.

[0137] like Figure 7The diagram shows a comparison of entropy error between the method of this embodiment and the array receiving information in a non-circular uniform linear array system. The simulation conditions are: number of snapshots J = 1, number of information sources K = 1, number of simulations T = 1000, the number of array elements in a typical nested array is N1 = 8 for dense subarrays and N2 = 9 for sparse subarrays; and the number of array elements in a uniform linear array is M = 17.

[0138] from Figure 3 and Figure 6 As can be seen from the data, the performance evaluation index for receiving DOA information in a non-circular sparse array in this embodiment verifies that the amount of DOA information received by a typical nested array in a sparse array exhibits two trends with increasing SNR. At low SNR, the amount of DOA information increases significantly with increasing SNR, indicating that the source information is effectively utilized, allowing for a rough estimation of the source's location. At high SNR, the increase in the amount of DOA information received by the array slows down, but the estimation accuracy gradually increases. At this point, the amount of DOA information shows a linear relationship with the logarithm of SNR, because the source's location is known, so the amount of DOA information no longer changes significantly with SNR. The entropy error also verifies this phenomenon and is inversely proportional to the amount of received DOA information. Furthermore, it can be observed that the amount of DOA information received by the array is directly proportional to the total number of array elements, N.

[0139] from Figure 4 As can be seen, when the total number of array elements N in a typical nested array is fixed, changing the number of array elements in the two subarrays and observing the relationship between the theoretical value of DOA information and SNR reveals that when the number of array elements in the two subarrays is 9 and 10 respectively, the array obtains more DOA information than arrays with other array element configurations. This is because, under the premise that the total number of array elements remains unchanged, the DOA information is maximized when the number of array elements in the two subarrays is equal. Therefore, the non-circular nested array system obtains the most DOA information, has the least uncertainty in source azimuth estimation, and has the best estimation performance.

[0140] from Figure 5 and Figure 7 As can be seen, the amount of DOA information obtained by the nested array is greater than that obtained by the uniform linear array throughout the entire SNR range. This demonstrates the superior DOA estimation performance of the nested array. Under the same array element configuration, sparsely arranging the array elements can capture more DOA information about the source from the received signal, making the parameter estimation more accurate and saving the hardware cost of the actual array construction.

[0141] In summary, the analysis of the simulation results shows that the method in this embodiment uses the probability density function of the parameter to be estimated to obtain the formula for DOA information content, and obtains an approximate upper bound of DOA information content under high signal-to-noise ratio conditions. Furthermore, a specific expression for CRB is given for non-circular nested array systems. Numerical simulation verifies that it is consistent with the approximate variance expression of PDF under high SNR conditions.

[0142] Meanwhile, the optimal array configuration of the non-circular sparse array was obtained by utilizing the DOA information. That is, under the premise of a fixed total number of array elements, the closer the number of array elements of two subarrays are, the better the DOA estimation performance. In addition, the use of sparse arrays also further saves the hardware cost of array construction. Finally, a DOA estimation performance evaluation index for non-circular sparse array systems, namely entropy error, is proposed to evaluate the information acquisition capability of the system. The rationality of the proposed index is verified by simulation and comparison with CRB.

[0143] In addition to the embodiments described above, the present invention may have other implementations. All technical solutions formed by equivalent substitution or equivalent transformation fall within the protection scope claimed by the present invention.

Claims

1. A method for evaluating the performance of DOA estimation for non-circular sparse matrices based on information theory, characterized in that: Includes the following steps S1. Construct a multidimensional probability density function using the received signal x(t) through a sparse array. The joint probability density function p(x,θ) of the sparse array received signal x(t) and the source azimuth angle θ is derived using information theory methods. S2. Derive the DOA posterior probability density function p(θ|x) of the noncircular sparse matrix based on the joint probability density function p(x,θ), and simplify it using Bessel functions; S3. Obtain the DOA information I(X; Θ) of the non-circular sparse array according to the mutual information formula; S4. Using the joint probability density function p(x,θ) of DOA, derive the posterior probability density function p(θ|n) of DOA under given noise conditions. Simplify using Taylor expansion, and obtain the information-theoretic approximate upper bound I of the non-circular sparse array DOA according to the differential entropy formula for variables following a Gaussian distribution. aub (X;Θ); S5. Using the posterior differential entropy h(Θ|X), we obtain an index for evaluating the DOA estimation performance of a non-circular sparse array system, namely, the entropy error. And derive the CRB of non-circular nested arrays, i.e. Will and By comparison, the performance evaluation of the non-circular sparse array system in the entire signal-to-noise ratio range is analyzed.

2. The performance evaluation method for DOA estimation of non-circular sparse matrices based on information theory according to claim 1, characterized in that: In step S1, the sparse array is a general nested array with a total number of array elements of N. The general nested array includes a dense subarray and a sparse subarray. The dense subarray has N1 array elements and the array element spacing is d0. The sparse subarray has N2 array elements and the array element spacing is (N1+1)d0. Where d0=λ / 2, λ is the wavelength, and the total number of array elements N=N1+N2.

3. The method for evaluating the performance of non-circular sparse matrix DOA estimation based on information theory according to claim 2, characterized in that: In step S1, it is assumed that a single far-field narrowband non-circular signal is incident on a general nested array with a direction of arrival θ in a certain space, and the received signal x(t) is defined as: x(t)=a(θ)s(t)+n(t) The non-circular signal is s(t) = Φs R (t), s R (t) represents a real-valued signal. Contains non-circular information, where j represents an imaginary number. θ represents the non-circular phase; n(t) is additive white Gaussian noise with zero mean and variance N0 on a general nested matrix; a(θ) is the direction vector of the general nested matrix; Since the noise n(t) is additive white Gaussian noise, at the source azimuth θ and in non-circular phase... Given the given conditions, the multidimensional probability density function of the array-received signal x is: in, The non-circular phase is represented by N, the total number of array elements is N, a(θ) is the direction vector of a general nested array, N0 is the variance of additive white Gaussian noise, and s and s R Let (·) represent a non-circular signal and a real-valued signal, respectively. H Indicates conjugate transpose; The joint probability density function of the array received signal x and the source azimuth θ is derived. in, Indicates a known non-circular phase Under the condition of , the probability density function of the array receiving signal x; Indicates non-circular phase The probability density function.

4. The performance evaluation method for DOA estimation of a non-circular sparse matrix based on information theory according to claim 1, characterized in that: In step S2, the non-circular sparse matrix DOA information θ and phase are... All follow a uniform distribution, so their probabilities are all constants; based on p(x,θ), the posterior probability density function of the DOA of the non-circular sparse matrix is ​​obtained as follows: Where ||Θ|| represents the length of the DOA observation interval; (·) H Denotes the conjugate transpose; x in the multidimensional probability density function of the array received signal x H The value of x depends on the actual value of the source DOA, so it is omitted in the above formula.

5. The performance evaluation method for DOA estimation of a non-circular sparse matrix based on information theory according to claim 4, characterized in that: In step S2, the numerator in the expression for the posterior probability density function of the DOA of a non-circular sparse matrix is ​​simplified to obtain: Where Im(·) represents the imaginary part of the complex number; substituting the above equation into the expression for the posterior probability density function of the DOA of a non-circular sparse matrix, we get: Where I0(·) represents the Bessel function.

6. The performance evaluation method for DOA estimation of a non-circular sparse matrix based on information theory according to claim 1, characterized in that: In step S3, the information content of DOA is obtained according to the formula for mutual information, which is the difference between the differential entropy of the prior and posterior distributions of DOA. Among them, E x [·] indicates the desired outcome.

7. The performance evaluation method for DOA estimation of a non-circular sparse matrix based on information theory according to claim 1, characterized in that: In step S4, the posterior probability density function p(θ|x) of DOA under given noise conditions is derived using the formula p(x,θ) for the posterior probability density function of DOA: Where I0(·) represents the Bessel function, (·) H This indicates taking the conjugate transpose, where θ0 is the assumed actual direction of the signal received by the array. Given non-circular phase information; Furthermore, the approximate upper bound of the DOA information content of non-circular nested matrices is obtained as follows: Where, σ 2 Let p(θ|n) be the variance of the posterior probability density function of DOA under given noise conditions, which follows a Gaussian distribution.

8. The performance evaluation method for DOA estimation of a non-circular sparse matrix based on information theory according to claim 1, characterized in that: In step S5, the posterior differential entropy h(Θ|X) represents the uncertainty of the unknown parameter Θ given the vector X, and is used to evaluate the estimation performance of the system. Therefore, the entropy error of the nested array is used as an indicator to accurately evaluate the estimation performance of the nested array system, and its specific expression is as follows: in, This represents the entropy error.

9. The performance evaluation method for DOA estimation of a non-circular sparse matrix based on information theory according to claim 8, characterized in that: In step S5, the known CRB of the unbiased estimator θ is: in: Based on the direction matrix of the nested matrix, the following expression is obtained: a H (θ)a(θ)=N Therefore, the DOA estimate CRB of the non-circular sparse matrix is ​​obtained as follows: in, Represents a non-circular nested array of CRBs.

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