A mixed-precision quantization-based coprime array direction of arrival estimation method

Abstract

The application discloses a kind of based on mixed precision quantification coprime array direction-of-arrival estimation method.The implementation steps are: arranging mixed precision extended coprime array;Construct a classic signal receiving model;Solve the ideal mixed precision signal model constructed;Construct finite snapshot mixed precision array signal model;Signal whitening model is constructed;Optimization problem for reconstructing mixed precision quantification theoretical covariance matrix is constructed and is solved.The advantages of the method: compared with full high-precision array, the array relieves the quantization pressure of ADC, reduces sampling complexity.Compared with full low-precision array, the array has higher estimation accuracy.Coprime array can obtain higher estimation accuracy and stronger resolution than uniform array, and has higher degree of freedom.On the other hand, the method restores the elements of the hole in the coprime array by kernel norm minimization, and adds the whitening method to eliminate the error caused by finite snapshots, with higher estimation accuracy.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of array signal processing, and particularly relates to a method for estimating direction of arrival of a coprime array based on mixed precision quantization. BACKGROUND

[0002] Array signal processing is widely used in mobile communication, satellite communication, radar and other technical fields. In the past, the ADCs used in array are high-precision, but the energy consumption and production cost of the ADC are exponentially related to the number of quantization bits. In order to alleviate the hardware pressure of the ADC, 1-bit quantization is proposed. Compared with high-precision ADC, 1-bit ADC is easy to implement in hardware and has low cost. Only the sign information of each measurement value is retained, the quantization pressure of the ADC is alleviated, and the sampling complexity is reduced. At the same time, using 1-bit ADC will also lead to the decline of the parameter estimation performance of the system.

[0003] In addition, the coprime array generalizes the second-order statistics of the received signal to the virtual domain, and obtains an array structure with more array elements in the virtual domain. The virtual array elements of the coprime array have holes, and the maximum continuous virtual array elements of the coprime array or the holes can be filled to utilize all the virtual array elements. Document 1 (Pal P., Vaidyanathan PP. Coprime sampling and the MUSIC algorithm [C]. 2011 Digital Signal Processing and Signal Processing Education Meeting, 2011: 289-294.). This method uses a spatial smoothing MUSIC algorithm, so it can only use the maximum virtual array elements of the coprime array, resulting in the loss of information at the position of the discontinuous array elements and the loss of degrees of freedom. Document 2 (C. Zhou, Y. Gu, Z. Shi and M. Haardt, "Direction-of-Arrival Estimation for Coprime Arrays via Coarray Correlation Reconstruction: A One-Bit Perspective," 2020 IEEE 11th Sensor Array and Multichannel Signal Processing Workshop (SAM), 2020, pp. 1-4). This method uses a 1-bit quantized coprime array, and fills the holes of the virtual array elements by using the kernel norm, so as to use all the virtual array elements of the coprime array. However, this method uses 1-bit ADC, resulting in a decline in the performance of the direction-of-arrival estimation. Document 3 (L. Wang, C. Ren and Z. Zheng, "DOA Estimation for Monostatic Coprime MIMO Radar With Mixed-Resolution Quantization," in IEEE Transactions on Vehicular Technology, vol. 72, no. 12, pp. 16737-16741, Dec. 2023). This method uses a mixed-precision coprime array, and fills the holes of the virtual array elements of the coprime array by using the atomic norm. However, the estimation accuracy of the atomic norm minimization is limited. SUMMARY

[0004] This invention addresses the shortcomings of existing technologies by proposing a direction-of-arrival (DOA) estimation method for coprime arrays based on mixed-precision quantization. This method processes signals received by a coprime array with mixed precision and performs DOA estimation, thereby effectively improving the accuracy of DOA estimation.

[0005] To solve the above-mentioned technical problems, the technical solution of the present invention is as follows:

[0006] A method for estimating the direction of arrival (DOA) of a coprime array based on mixed-precision quantization includes the following steps:

[0007] (1) Deploying a hybrid precision extended coprime array: This array consists of two subarrays. Assume the first subarray consists of 2M1 elements, with an interval of M2d between adjacent elements; the second subarray consists of M2 elements, with an interval of 2M1d between adjacent elements. M1 and M2 are coprime, and M1 < M2, d = λ / 2, where λ represents the signal wavelength. The set of element positions in this coprime array can be represented as:

[0008]

[0009] Where ∪ represents the merge operation. The array has a total of M = 2M1 + M2 - 1 elements. The first subarray uses a high-precision ADC, the second subarray uses a low-precision ADC, and since the element at position 0 is shared by both subarrays, the element at position 0 uses a high-precision ADC.

[0010] (2) Constructing a classic signal reception model: Assuming the signal comes from θ = [θ1, θ2, ..., θ3] K ] T When K far-field narrowband uncorrelated signals in the direction are incident on the array, the signal received by the coprime array at time t can be expressed as:

[0011] x(t) = As(t) + n(t)

[0012] Where s(t)∈ K×1 This represents the signal vector at time t. Let represent the additional Gaussian white noise vector, and let the noise power be . A=[a(θ1),a(θ2),,a(θ K )]∈ M×K The manifold matrix of the array is represented by, This represents the steering vector of the array for the k-th received signal. (·) T This indicates the transpose operation.

[0013] The covariance matrix corresponding to x(t) is expressed as:

[0014]

[0015] where, denotes the power of the signal, (·) H denotes the conjugate transpose operation.

[0016] The vectorization operation on the covariance matrix can be obtained as:

[0017]

[0018] In the equation, 1 m = vec(I M ), e denotes the Khatri-Rao product, and vec denotes the vectorization operation. The matrix A * ⊙ A can be regarded as the array flow matrix of the differential virtual array, and its sensor position can be represented as:

[0019]

[0020] The number of elements in the set is defined as D. The elements corresponding to the repeated elements in the vector r are averaged, and the remaining elements are reordered to obtain a new vector It can be obtained by averaging the rows corresponding to the repeated elements of the virtual array elements in A * ⊙ A and then repeating the sorting. 1' represents a column vector composed of 0 and 1.

[0021] (3) Constructing an ideal mixed-precision signal model: the first subarray uses a high-precision ADC, and the second subarray uses a low-precision ADC. The output of the first subarray is represented as:

[0022] y H (t) = A H s(t) + n H (t)

[0023] where A H denotes the array flow matrix of the first subarray, and n H (t) denotes the corresponding additive white Gaussian noise. The output of the second subarray is represented as:

[0024] y L (t) = αx L (t) + n q (t) = αA L s(t) + αn L (t) + n q (t)

[0025] where A L denotes the array flow matrix of the second subarray, and n L(t) represents the corresponding additive white Gaussian noise, n q (t) represents the quantization noise, whose mean is 0, and the covariance matrix can be represented as α represents the linear quantization gain, and ρ represents the distortion coefficient, the relationship between them can be represented as α = 1-ρ, which can be obtained by looking up the table.

[0026] In summary, the output of the mixed-precision array can be represented as:

[0027]

[0028] wherein, is a matrix with only diagonal elements, each element on the diagonal corresponds to the same row of the stream matrix. Therefore, the position corresponding to the first stream matrix is set to 1, and the position corresponding to the second stream matrix is set to α. is a matrix composed of n q (t) rows, and the rows corresponding to the positions of the two stream matrices are n q (t), and the remaining positions are 0.

[0029] The covariance matrix of y(t) can be represented as:

[0030]

[0031] Vectorize R y to get a new vector wherein q is a vector that satisfies wherein represents the dot product of the matrix, represents the Kronecker product.

[0032] Average the corresponding elements of the repeated virtual array elements in , and reorder the remaining elements to get:

[0033]

[0034] Because there are holes in the virtual array elements of the coprime array, in order to utilize all the virtual array elements, the holes must be filled. Assuming that ideal interpolation is performed, the above formula can be rewritten as:

[0035]

[0036] wherein The number of elements in the set is defined as V. Based on the above vector, a LxL Toeplitz matrix is constructed:

[0037]

[0038] where L = (V + 1) / 2. The above matrix can be further written as Q is a matrix containing quantization information,

[0039] (4) Constructing a finite snapshot number mixed-precision array signal model: In practice, only the sample covariance matrix R can be obtained:

[0040]

[0041] Next, the vectorization operation is performed on the matrix, followed by averaging the corresponding elements of the repeated virtual elements, and reordering the remaining elements to obtain the vector Next, the zero insertion preprocessing operation is performed on the holes of the vector to obtain

[0042]

[0043] Based on the above vector, a L x L Toeplitz matrix is constructed:

[0044]

[0045] (5) Constructing a signal whitening model: Due to the limited number of snapshots, there is an error between the sample covariance matrix and the covariance matrix and satisfies the following distribution:

[0046]

[0047] where represents an asymptotic complex normal distribution with mean 0 and covariance matrix Σ, Because the sample covariance matrix is vectorized, the repeated element average processing, zero insertion at the hole, and finally spliced into a Toeplitz matrix, the solution of the corresponding covariance matrix is required. First, the repeated virtual element corresponding elements in vec(ΔR) are averaged Δz = G * vec(ΔR), where is a binary matrix. Next, the zero insertion preprocessing is performed on the holes where is a binary matrix. Finally where is a binary matrix. Therefore, we can get:

[0048]

[0049] (6) Constructing an optimization problem for reconstructing the mixed-precision quantization theoretical covariance matrix and solving:

[0050]

[0051] wherein, ε represents a threshold value for constraining fitting error, rank represents the rank of the matrix, ||·||2 represents the two norm. This formula is an NP-hard problem, which can be converted into the following formula for solving:

[0052]

[0053] subject to R≥0 wherein, μ is a regularization parameter, tr(·) represents the trace of the matrix, F is a binary matrix corresponding to The positions with values are set to 1, and the rest are set to 0, T=(G*H*J*Σ*J H *H H *G H )^(-0.5). Finally, the DOA estimation value is obtained by the MUSIC algorithm.

[0054] As preferred, the mixed-precision extended array in step one can be set as a high-precision subarray and a low-precision subarray.

[0055] As preferred, the array needs to be interpolated in step three, and if the array is changed to a nested array, interpolation is not needed.

[0056] As preferred, the vector needs to be spliced into a Toeplitz matrix in step three, and a Toeplitz matrix can be directly generated according to the vector.

[0057] As preferred, the matrix rank minimization in step six can be converted into the trace of the matrix or the kernel norm of the matrix.

[0058] The present application has the following characteristics and beneficial effects:

[0059] By adopting the above technical solution, firstly, the mixed-precision quantization-based array wave direction estimation method proposed by the present application adopts a mixed-precision quantization array, which relieves the quantization pressure of the ADC and reduces the sampling complexity compared with a full high-precision array. Compared with a full low-precision array, the array has higher estimation accuracy. Compared with a uniform array, the array can obtain higher estimation accuracy and stronger resolution and has higher degrees of freedom. On the other hand, the method restores the elements of the holes in the array by kernel norm minimization and adds a whitening method to eliminate the error caused by limited snapshots, and has higher estimation accuracy. BRIEF DESCRIPTION OF DRAWINGS

[0060] In order to make the technical solutions in the embodiments of the present application or the prior art clearer, the accompanying drawings needed in the embodiments or prior art description will be briefly introduced. Obviously, the accompanying drawings in the following description only show some embodiments of the present application, and other drawings can be obtained by those skilled in the art without any creative effort based on these drawings.

[0061] Figure 1 is a general flow chart of the method of the present application;

[0062] Figure 2 is a table of distortion coefficients corresponding to different quantization precisions in the present application;

[0063] Figure 3 is the arrangement of the mixed precision array in the present application;

[0064] Figure 4 is a comparison diagram of the estimation accuracy of the method of the present application and several algorithms under different signal-to-noise ratios;

[0065] Figure 5 is a comparison diagram of the estimation accuracy of the method of the present application and several algorithms under different snapshot numbers;

[0066] Figure 6 is a comparison diagram of the estimation accuracy of the method of the present application and low-precision and high-precision arrays under different signal-to-noise ratios. DETAILED DESCRIPTION

[0067] It should be noted that the embodiments in the present application and the features in the embodiments can be combined with each other without conflict.

[0068] In order to make the purpose, technical solutions and advantages of the present application clearer, the present application will be further described in detail below with reference to the accompanying drawings.

[0069] On the contrary, the present application covers any substitution, modification, equivalent method and solution made on the essence and scope of the present application defined by the claims. Further, in order to make the public have a better understanding of the present application, some specific details are described in the following detailed description of the present application. The present application can also be completely understood without the description of these details by those skilled in the art.

[0070] Embodiment 1

[0071] The present application provides a coprime array direction of arrival estimation method based on mixed precision quantization, as shown in Figure 1 The method comprises the following steps:

[0072] Step 1: Deploy a hybrid precision extended coprime array: This array consists of two subarrays. Assume the first subarray consists of 2M1 elements, with an interval of M2d between adjacent elements; the second subarray consists of M2 elements, with an interval of 2M1d between adjacent elements. M1 and M2 are coprime, and M1 < M2, d = λ / 2, where λ represents the signal wavelength. The set of element positions in this coprime array can be represented as:

[0073]

[0074] Where ∪ represents the merge operation. The array has a total of M = 2M1 + M2 - 1 elements. The first subarray uses a high-precision ADC, the second subarray uses a low-precision ADC, and since the element at position 0 is shared by both subarrays, the element at position 0 uses a high-precision ADC.

[0075] Step 2: Construct a classic signal reception model: Assume the signal comes from θ = [θ1, θ2, ..., θ3]. K ] T K far-field narrowband uncorrelated signals in the direction are incident on the array. The signal received by the coprime array at time t can be expressed as:

[0076] x(t) = As(t) + n(t)

[0077] in This represents the signal vector at time t. Let represent the additional Gaussian white noise vector, and let the noise power be . The manifold matrix of the array is represented by, This represents the steering vector of the array for the k-th received signal. (·) T This indicates the transpose operation.

[0078] The covariance matrix corresponding to x(t) is expressed as:

[0079]

[0080] in, Indicates the power of the signal, (·) H This indicates the conjugate transpose operation.

[0081] By vectorizing the covariance matrix, we can obtain:

[0082]

[0083] In the formula, 1 m =vec(I M), e denotes Khatri-Rao product, and vec denotes vectorization operation. Matrix A * A can be regarded as the array flow matrix of the differential virtual array, and the sensor positions can be expressed as:

[0084]

[0085] The number of elements in the set is defined as D. The elements corresponding to the repeated elements in the vector r are averaged, and the remaining elements are reordered to obtain a new vector A can be obtained by averaging the rows corresponding to the repeated elements of the virtual elements in A * A. 1' denotes a column vector consisting of 0 and 1.

[0086] Step three: Constructing the ideal mixed-precision signal model: the first subarray uses high-precision (regular-precision) ADC, and the second subarray uses low-precision (1-bit) ADC. The output of the first subarray is expressed as:

[0087] y H (t) = A H s(t) + n H (t)

[0088] where A H denotes the array flow matrix of the first subarray, n H (t) denotes the corresponding additive white Gaussian noise. The output of the second subarray is expressed as:

[0089] y L (t) = αx L (t) + n q (t) = αA L s(t) + αn L (t) + n q (t)

[0090] where A L denotes the array flow matrix of the second subarray, n L (t) denotes the corresponding additive white Gaussian noise, and n q (t) denotes the quantization noise, whose mean is 0, and the covariance matrix can be expressed as α denotes the linear quantization gain, and ρ denotes the distortion coefficient, and their relationship can be expressed as α = 1 - ρ, as shown in Figure 2 .

[0091] In summary, the output of the mixed-precision array can be expressed as:

[0092]

[0093] where, is a matrix with only diagonal elements, each of which corresponds to the same row of the streamer matrix. Thus, the positions corresponding to the first streamer matrix are set to 1, and the positions corresponding to the second streamer matrix are set to a. is a matrix composed of n q (t) rows, each of which corresponds to the position of the streamer matrix. The rows corresponding to the positions of the two streamer matrices are n q (t), and the remaining positions are 0.

[0094] The covariance matrix of y(t) can be expressed as:

[0095]

[0096] Vectorize R y to obtain a new vector where q is a vector that satisfies where denotes the dot product of matrices, denotes the Kronecker product.

[0097] Average the corresponding elements of the repeated virtual array elements in , and reorder the remaining elements to obtain:

[0098]

[0099] Because there are holes in the virtual array elements of the coprime array, in order to utilize all virtual array elements, the holes must be filled. Assuming that ideal interpolation is performed, the above equation can be rewritten as:

[0100]

[0101] where The number of elements in this set is defined as V. Based on the above vector, a LxL Toeplitz matrix is constructed:

[0102]

[0103] where L = (V+1) / 2. The above matrix can be further written as Q is a matrix containing quantization information,

[0104] Step four: Construct a finite snapshot number hybrid precision array signal model: In practice, only the sample covariance matrix can be obtained:

[0105]

[0106] Next, the matrix is vectorized, then the repeated virtual elements are averaged, and the remaining elements are reordered to obtain the vector Next, the vector is preprocessed by inserting zeros at the holes to obtain

[0107]

[0108] Based on the above vector, an LxL Toeplitz matrix is constructed:

[0109]

[0110] Step five: Construct the signal whitening model: Because of the limited number of snapshots, there is an error between the sample covariance matrix and the covariance matrix And satisfies the following distribution:

[0111]

[0112] where, represents the asymptotic complex normal distribution with mean 0 and covariance matrix Σ, Because the sample covariance matrix is vectorized, the repeated elements are averaged, the holes are filled with zeros, and finally a Toeplitz matrix is constructed, so we need to solve the corresponding covariance matrix. First, the repeated virtual elements in vec(ΔR) are averaged Δz = G * vec(ΔR), where is a binary matrix. Then, the holes are preprocessed by filling zeros where is a binary matrix. Finally where is a binary matrix. Therefore, we can get:

[0113]

[0114] Step six: Construct an optimization problem for reconstructing the mixed-precision quantization theoretical covariance matrix and solve it:

[0115]

[0116] where ε represents a threshold for constraining the fitting error, rank represents the rank of the matrix, and ||·||2 represents the two-norm. This formula is an NP-hard problem, which can be transformed into the following formula to solve:

[0117]

[0118] subject to R≥0 where μ is a regularization parameter, tr(·) denotes the trace of a matrix, F is a binary matrix corresponding to with the positions of the values set to 1 and the rest set to 0, T = (G*H*J*Σ*J H *H H *G H )^(-0.5). Finally, the DOA estimate is obtained by the MUSIC algorithm.

[0119] Example 2

[0120] The difference between this example and Example 1 is that the number of array elements M = 2M1+M2-1 of the mixed-precision coprime array in this example is set to 10, M1= 3, and M2= 5, and the specific arrangement is as shown in Table 1 (where the low-precision array uses a 1-bit ADC). The number of signal sources K is 2. The angles of incidence of the two signals are -30° and 40°, the minimum interval between array elements d = λ / 2, and the regularization parameter λ = 0.25. The signal-to-noise ratio is set to change from -10 dB to 25 dB at intervals of 5 dB. The number of sampling snapshots is set to 100. Figure 3

[0121] The root mean square error of the DOA estimation of the method in this example under different signal-to-noise ratios is compared with the estimation accuracy of the atomic norm, the kernel norm, SSMUSIC, and the mixed-precision Cramer-Rao bound, and the comparison results are shown in Table 2. As can be seen from the table, compared with the other algorithms, the root mean square error of this algorithm is the lowest, and the estimation performance is very good. Figure 4

[0122] Specifically, the comparative algorithms refer to references 2-4. Subsequently, the signal-to-noise ratio is fixed at 20 dB, and the number of snapshots is changed from 30 to 300 at intervals of 30. Other parameters remain unchanged, and the root mean square error of the DOA estimation of the method in this example under different numbers of snapshots is compared with the estimation accuracy of the atomic norm, the kernel norm, SSMUSIC, and the mixed-precision Cramer-Rao bound, and the comparison results are shown in Table 3. As can be seen from the table, compared with the other algorithms, the root mean square error of this algorithm is the lowest, and the estimation performance is very good. Figure 5

[0123] It should be noted that all the statistical results of this example and the comparative examples are based on 500 times of Monte Carlo experiments.

[0124] Example 3

[0125] ​​​The difference between the embodiment and the embodiment 1 is that the performance of the embodiment is stronger than the low-precision performance with the change of the signal-to-noise ratio. The incident angles of the two signals are -4° and 5°, the signal-to-noise ratio is set to change from -15 dB to 20 dB with an interval of 5 dB. The sampling snapshot number is set to 200, the remaining parameters are unchanged, the root mean square error of the direction of arrival estimation of the method in the different snapshot numbers is compared with the estimation accuracy of the full high-precision array and the full low-precision array, and the comparison result is shown in the following table. Figure 6 As shown in the figure, compared with the low-precision array, the performance of the algorithm is better, and the estimation error is not very large compared with the full high-precision array.

[0126] The embodiments of the present application are described in detail above with reference to the drawings, but the present application is not limited to the described embodiments. For those skilled in the art, various changes, modifications, replacements and variations of the embodiments including components are made without departing from the principles and spirits of the present application, and still fall within the protection scope of the present application.

Claims

1. A method for estimating the direction of arrival (DOA) of a coprime array based on mixed-precision quantization, characterized in that, Includes the following steps: Step 1: Arrange a mixed-precision extended coprime array, wherein the coprime array comprises two subarrays; Step 2: Define from θ = [θ1, θ2, ..., θ3] K ] T K far-field narrowband uncorrelated signals in the direction are incident on a coprime array; Step 3: Calculate the covariance matrix corresponding to the signal x(t) received by the coprime array at time t, and perform vectorization operation on the covariance matrix to obtain vector r. Average the elements of the corresponding repeating elements in vector r, and reorder the remaining elements. Step 4: The first subarray in the coprime array uses a high-precision ADC, and the second subarray uses a low-precision ADC, thus obtaining a mixed-precision array; Step 5: Calculate the covariance matrix R of the mixed-precision array. y And for the covariance matrix R y Perform vectorization operations to obtain vectors right The elements corresponding to the repeated virtual array elements are averaged, and the remaining elements are reordered. The holes are filled by one-time ideal interpolation for the reordered virtual array elements. An L×L Toplitz matrix is ​​constructed based on the filled virtual array elements. Step 6: Obtain the sample covariance matrix using a mixed-precision array with a finite number of snapshots. Vectorize the sample covariance matrix, then average the corresponding elements of the repeated virtual matrix elements, and reorder the remaining elements to obtain the vector. Next, a zero-insertion preprocessing operation is performed at the holes in the vector to obtain... According to the vector Construct another L×L Toplitz matrix; Step 7: Combine the sample covariance matrix and the covariance matrix R. y The difference is used to obtain the error. And it satisfies the following distribution: in, This represents an asymptotically complex normal distribution with a mean of 0 and a covariance matrix of Σ. Step 8: Average the elements corresponding to the repeated virtual array elements in vec(ΔR), and then fill the holes with zeros as a preprocessing step. Step 9: Based on the two L×L Toplitz matrices constructed in Steps 5 and 6, construct an optimization problem for reconstructing the covariance matrix of mixed precision quantization theory and obtain the DOA estimate using the MUSIC algorithm.

2. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 1, characterized in that, The first subarray consists of 2M1 array elements, with an interval of M2d between adjacent array elements; the second subarray consists of M2 array elements, with an interval of 2M1d between adjacent array elements; wherein M1 and M2 satisfy the coprime relationship, and M1 < M2, d = λ / 2, where λ represents the signal wavelength.

3. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 2, characterized in that, The set of positions of the array elements in the coprime array is represented as follows: Where ∪ represents the merge operation, the array has a total of M = 2M1 + M2 - 1 array elements.

4. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 3, characterized in that, In step 2, the signal received by the coprime array at time t is represented as follows: x(t) = As(t) + n(t) in This represents the signal vector at time t. Let represent the additional Gaussian white noise vector, and let the noise power be . Denotes the manifold matrix of the array, where, This represents the steering vector of the array for the k-th received signal, where, (·) T This indicates the transpose operation.

5. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 4, characterized in that, The covariance matrix corresponding to x(t) is expressed as: in, Indicates the power of the signal, (·) H This indicates the conjugate transpose operation.

6. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 5, characterized in that, In step 3, the covariance matrix corresponding to x(t) is vectorized to obtain: In the formula, 1 m =vec(I M ), ⊙ denotes the Khatri-Rao product, vec denotes the vectorization operation, where matrix A * ⊙A is the array manifold matrix of this differential virtual array; The elements of the corresponding repeating elements in vector r are averaged, and the remaining elements are reordered to obtain a new vector. By analyzing A * In ⊙A, the rows corresponding to the repeated virtual array elements are averaged and then sorted repeatedly to obtain 1', which represents a column vector composed of 0 and 1.

7. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 6, characterized in that, The matrix A * The position of sensor A is represented as follows: The number of elements in this set is defined as D.

8. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 7, characterized in that, In step 4, the output of the first subarray is represented as: y H (t)=A H s(t)+n H (t) Among them, A H Let n represent the array manifold matrix of the first subarray. H (t) represents the corresponding additive white Gaussian noise; The output of the second subarray is represented as: y L (t)=αx L (t)+n q (t)=αA L s(t)+αn L (t)+n q (t) Among them, A L Let n represent the array manifold matrix of the second subarray. L (t) represents the corresponding additive white Gaussian noise, n q (t) represents the quantization noise, and the covariance matrix is ​​expressed as... α represents the linear quantization gain, and ρ represents the distortion coefficient; The output of the mixed-precision array is represented as follows: in, It is a matrix with elements only on its diagonal, where each element on the diagonal corresponds to the same row of the manifold matrix.

9. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 8, characterized in that, The shared array element position of the first subarray and the second subarray is 0, therefore the array element at position 0 uses a high-precision ADC.

10. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 8, characterized in that, In the mixed precision array, the positions corresponding to the first manifold matrix are all set to 1, and the positions corresponding to the second manifold matrix are all set to α. It is composed of n q The matrix composed of (t) has n rows corresponding to the positions of the two manifold matrices. q (t), with the remaining positions all being 0.

11. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 8, characterized in that, The covariance matrix of y(t) is expressed as:

12. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 10, characterized in that, In step 5, R y Vectorize to obtain a new vector. Where q is a vector, satisfying in Represents the dot product of matrices. Represents the Kronecker product. right The elements corresponding to the repeated virtual matrix elements are averaged, and the remaining elements are reordered to obtain: For virtual elements of a coprime array with holes, to fill these holes, assuming an ideal interpolation, the above formula can be rewritten as: in The number of elements in this set is defined as V. Based on the vector above, construct an L×L Toplitz matrix: Where L = (V+1) / 2, the matrix above can be further written as Q is a matrix containing quantitative information.

13. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 12, characterized in that, The sample covariance matrix is:

14. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 13, characterized in that, In step 6, the sample covariance matrix is ​​vectorized, then the elements corresponding to the repeated virtual matrix elements are averaged, and the remaining elements are reordered to obtain a vector. Next, a zero-insertion preprocessing operation is performed at the holes in the vector to obtain... Based on the vectors above, construct an L×L Toplitz matrix:

15. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 14, characterized in that, In step 7, firstly, the elements corresponding to the repeated virtual array elements in vec(ΔR) are averaged, Δz = G*vec(ΔR), where... It is a binary matrix, and then the holes are preprocessed by padding with zeros. in It is a binary matrix, and finally in It is a binary matrix, therefore we get:

16. The method for estimating the direction of arrival of coprime arrays based on hybrid precision quantization according to claim 15, characterized in that, In step 9, the constructed optimization problem expression is as follows: Where ε represents a threshold used to constrain the fitting error, rank represents the rank of the matrix, and ||·||2 represents the L2 norm. This expression is an NP-hard problem, which can be solved by solving the following expression: subject to R≥0 Where μ is the regularization parameter, tr(·) represents the trace of the matrix, and F is a binary matrix, corresponding to All positions with values ​​are set to 1, and the rest are set to 0. T = (G*H*J*Σ*J) H *H H *G H ) -0.5 .

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