A method for estimating the direction of arrival of a wave in the presence of calibration errors in an array sensor

Abstract

The application discloses a method for estimating the direction of arrival under the calibration error of an array sensor, which comprises the following steps: step 1: according to the sampling rule of the incident signal, selecting the time domain sampling point number T, establishing the matrix system Y=B·S+E+N of the array sampling under the calibration error of the sensor, and taking the sampling point number T as the column number of the left-end item matrix, wherein the matrix Y is an array sampling matrix of MxT, M is the number of array sensors, the matrix B is an array manifold matrix of MxK, K is the number of signal sources, the matrix S is a signal matrix of KxT, the matrix E is a calibration error matrix of MxT, and the matrix N is a noise matrix of MxT; step 2: according to a preset rule, constructing an over-complete dictionary matrix A of the array manifold matrix, and obtaining the matrix system Y=A·X+E+N of the array sampling under the sparse representation, wherein the matrix A is an over-complete dictionary matrix of MxN, N is the column number of the dictionary, and the matrix X is a signal sparse representation matrix of NxT; step 3: under the low-rank and row sparse decomposition framework, constructing a convex optimization formula according to the matrix system Y=A·X+E+N of the array sampling under the sparse representation; step 4: solving the constructed convex optimization formula by using the iterative reweighting method to obtain the solution matrix and ; and step 5: obtaining the corresponding direction of arrival estimation and the calibration error sensor position according to the non-zero row positions of the solution matrix and. The application can calculate two solution matrices and by using the iterative reweighting method, and obtain the direction of arrival estimation and the calibration error sensor position by using the solution matrix and respectively.

Description

Technical Field

[0001] This invention relates to the field of wave direction estimation technology, and more specifically to a method for wave direction estimation under array sensor calibration errors. Background Technology

[0002] In studies on direction-of-arrival estimation, compressed sensing technology is often used to solve this type of problem. The process of solving the direction-of-arrival estimation problem using compressed sensing technology is as follows: First, the matrix form of the array sampling data can be represented as:

[0003] Y = B·S + N

[0004] Wherein, matrix Y is an M×T array sampling matrix, M is the number of array sensors, T is the number of sampling points, matrix B is an M×K array manifold matrix, K is the number of uncorrelated signal sources, matrix S is a K×T signal matrix, and matrix N is an M×T noise matrix.

[0005] Based on the spatial partitioning, the overcomplete dictionary matrix A of the array manifold matrix is ​​constructed, and the array sampling matrix is ​​rewritten as:

[0006] Y = A·X + N

[0007] Where matrix A is an M×N overcomplete dictionary matrix, and matrix X is an N×T sparse representation matrix of the signal, where each non-zero row is equal to the row at the corresponding position in S. Based on the array sampling matrix under the sparse representation, the following convex optimization formula can be constructed to recover X:

[0008]

[0009] Among them ||·|| F Denotes the F norm of a matrix, ||·|| 2，1 This means taking the 2-norm of a vector in each row of the matrix and then taking the 1-norm of the resulting vector. This is achieved by solving the matrix. The position of the non-zero row can be used to estimate the direction of incoming waves.

[0010] However, the above methods will not work when the array sensors have calibration errors. Current solutions to this problem include maximum likelihood estimation, but this method requires the number of correctly calibrated sensors to exceed the number of signals; rotation-invariant techniques are also used to jointly estimate the direction of arrival, sensor gain, and phase in a uniform linear array; however, this method requires prior knowledge of the location of the calibrated sensor and other relevant information; the maximum entropy criterion has also been used to develop an arrival estimation method for cases where the location of the calibrated sensor is unknown; however, this method cannot detect the location of the calibrated sensor.

[0011] A solution is proposed to simultaneously obtain the direction of arrival and the location of the calibrated sensor when the location and other relevant information of the calibrated sensor are unknown. The details are as follows: First, the matrix form of the array sampling data in the case of array sensor calibration error can be represented as:

[0012] Y = B·S + E + N

[0013] Where matrix E is the M×T calibration error matrix. Based on the spatial partitioning, an overcomplete dictionary matrix A of the array manifold matrix is ​​constructed, yielding the sparse representation of the array sampling matrix under the condition of array sensor calibration errors, as follows:

[0014] Y = A·X + E + N

[0015] In this matrix, each non-zero row of matrix X equals the row corresponding to the position in matrix S, and each non-zero row of matrix E corresponds to the position of the sensor with a calibration error. Based on the sparse representation of the array sampling matrix, the following convex optimization formula can be constructed:

[0016]

[0017] Where μ is the smoothing coefficient, λ1 and λ2 are weighting coefficients, and 1 N Let N be a column vector of all 1s. M Let M be a column vector of all 1s. The solution matrix can be obtained using the iterative reweighting method. and By solving the matrix and The position of the non-zero row can be used to estimate the incoming wave direction and obtain the position of the calibrated error sensor within the same system framework. Summary of the Invention

[0018] The purpose of this invention is to provide a method for estimating the direction of arrival of an array sensor under calibration errors. The method involves constructing an overcomplete representation of the array manifold matrix to obtain a sparse representation of the array sampling matrix; using an iterative reweighting method to quickly solve for a row-sparse solution matrix; and finally estimating the direction of arrival and obtaining the location of the calibrated sensor within the same system framework by indexing the non-zero row positions of the solution matrix.

[0019] To achieve the above objectives, in combination Figure 1 , Figure 2 This invention proposes a method for estimating the direction of arrival under array sensor calibration errors, the method comprising:

[0020] Step 1: According to the incident signal sampling rules, select the number of time-domain sampling points T, and establish the array sampling matrix system Y = B·S + E + N under the condition of sensor calibration error. Take the number of sampling points T as the column number of the left-hand side matrix, where matrix Y is an M×T array sampling matrix, M is the number of array sensors, matrix B is an M×K array manifold matrix, K is the number of signal sources, matrix S is a K×T signal matrix, matrix E is an M×T calibration error matrix, and matrix N is an M×T noise matrix.

[0021] Step 2: Construct dictionary A according to the preset rules to obtain the matrix system Y = A·X + E + N for array sampling under sparse representation, where matrix A is an M×N overcomplete dictionary matrix, N is the number of columns in the dictionary, and matrix X is an N×T sparse representation matrix of the signal.

[0022] Step 3: Under the low-rank and row sparse decomposition framework, construct a convex optimization formula based on the matrix system Y = A·X + E + N of array sampling under sparse representation;

[0023] Step 4: Solve the convex optimization formula using the iterative reweighting method to obtain the solution matrices X and E;

[0024] Step 5: Based on the non-zero row positions of the solution matrices X and E, obtain the corresponding incoming wave direction estimates and the calibration error sensor positions.

[0025] In a further embodiment, the method further includes:

[0026] The incident direction is set to vary along the same direction, and the zenith angle θ and azimuth angle are used in the spherical coordinate system. This indicates the incident direction, where the zenith angle θ ranges from -90 to 90 degrees, and the azimuth angle... The range is 0-360 degrees.

[0027] In a further embodiment, in step S2, constructing a dictionary A according to preset rules to obtain the sparse representation matrix system Y = A·X + E + N means:

[0028] Based on a certain spatial interval, the zenith angle θ and azimuth angle are... Sampling is performed, and an array manifold matrix with an overcomplete dictionary A is constructed based on the sampled points.

[0029] In a further embodiment, the convex optimization formula in step S3 is as follows:

[0030]

[0031] In a further embodiment, the iterative reweighting method in step S4 includes the following steps:

[0032] S401: Set the smoothing coefficient μ, the tradeoff coefficients λ1 and λ2, and the maximum number of iterations I. max and threshold ε.

[0033] S402: Initial iteration number i = 0, initialize solution matrices X and E, denoted as X 0 and E 0 .

[0034] S403: Calculate the transition matrix P for the i-th iteration. i and Q i The specific calculation method is as follows:

[0035]

[0036] Where ||·||2 represents the vector 2-norm, and diag{·} represents constructing a diagonal matrix using the elements within {}, X i-1 (m) and E i-1 (m) represents the nth row of X and the mth row of E obtained from the (i-1)th iteration update, where n = 1, 2, ..., N, and m = 1, 2, ..., M.

[0037] S404: Calculate the solution matrices X and E for the i-th iteration. The specific calculation method is as follows:

[0038] X i =(A H ·A+λ1P i ) -1 ·A H ·(YE i-1 )

[0039] E i =(I+λ2Q) i ) -1 ·[YA·X i ]

[0040] in(·) H This represents the matrix conjugate transpose operation, (·). -1 This represents the matrix inversion operation.

[0041] S405: Calculate the relative error. The specific calculation method is as follows:

[0042]

[0043] Where f is the objective function, and the specific calculation method is as follows:

[0044]

[0045] After the calculation is complete, update the iteration count i = i + 1.

[0046] S406: Repeat steps S403 to S405 until i is greater than I. max Or the relative error Δ is less than ε.

[0047] Compared with existing technologies, the significant advantages of the above-described technical solution of the present invention are as follows:

[0048] (1) By introducing the sensor calibration error signal into the array sampling system model, the estimation of the incoming wave direction and the acquisition of the location of the calibration error sensor can be performed within the same system framework.

[0049] (2) By constructing a sparse representation of the array sampling matrix under the condition of array sensor calibration error, both the signal matrix and the calibration error matrix have row sparsity characteristics, which guarantees the successful inversion of the solution matrix. This allows the signal arrival direction and the position of the array calibration error sensor to be obtained without any prior information about the array and signal source.

[0050] (3) The solution matrix can be quickly calculated by the iterative reweighting method. Based on the non-zero rows of the two solution matrices, the corresponding incoming wave direction estimate and the position of the calibrated faulty sensor can be obtained respectively. Attached Figure Description

[0051] The accompanying drawings are not intended to be drawn to scale. In the drawings, each identical or nearly identical component shown in the various figures may be denoted by the same reference numeral. For clarity, not every component is labeled in each figure. Embodiments of various aspects of the invention will now be described by way of example and with reference to the accompanying drawings, wherein:

[0052] Figure 1 This is a flowchart of the method for estimating the direction of arrival of an array sensor under calibration errors according to the present invention;

[0053] Figure 2 This is a flowchart of the iterative reweighted least squares method of the present invention;

[0054] Figure 3 This is the column power diagram obtained by taking the vector 2 norm of each row of the solution matrices X and E. Detailed Implementation

[0055] To better understand the technical content of this invention, specific embodiments are described below in conjunction with the accompanying drawings. The purpose of this invention is to provide a method for estimating the direction of arrival (ROA) under calibration errors of an array sensor. This method obtains a sparse representation of the array sampling matrix by constructing an overcomplete representation of the array manifold matrix; it quickly solves for a row-sparse solution matrix using an iterative reweighting method; and by indexing the non-zero rows of the solution matrix, it achieves both the ROA estimation and the location of the calibrated sensor within the same framework.

[0056] To achieve the above objectives, in combination Figure 1 , Figure 2 This invention proposes a method for estimating the direction of arrival under array sensor calibration errors, the method comprising:

[0057] Step 1: According to the incident signal sampling rules, select the number of time-domain sampling points T, and establish the array sampling matrix system Y = B·S + E + N under the condition of sensor calibration error. Take the number of sampling points T as the column number of the left-hand side matrix, where matrix Y is an M×T array sampling matrix, M is the number of array sensors, matrix B is an M×K array manifold matrix, K is the number of signal sources, matrix S is a K×T signal matrix, matrix E is an M×T calibration error matrix, and matrix N is an M×T noise matrix.

[0058] Step 2: Construct dictionary A according to the preset rules to obtain the matrix system Y = A·X + E + N under sparse representation, where matrix A is an M×N dictionary, N is the number of columns in the dictionary, and matrix X is an N×T sparse signal representation;

[0059] Step 3: Under the low-rank and row sparse decomposition framework, construct a convex optimization formula based on the matrix system Y = A·X + E + N of array sampling under sparse representation;

[0060] Step 4: Solve the convex optimization formula using the iterative reweighting method to obtain the solution matrices X and E;

[0061] Step 5: Based on the non-zero row positions of the solution matrices X and E, obtain the corresponding incoming wave direction estimates and the calibration error sensor positions.

[0062] In a specific application embodiment, the detailed steps of S1 are as follows:

[0063] S101: Assume the array is a linear array containing 15 sensors, with a spacing of λ / 2 between two sensors, where λ is the operating wavelength. The signal incident direction varies only along the θ direction, ranging from [-90°, 90°]. Four uncorrelated signal sources are selected, with incoming wave directions of θ1 = -60°, θ2 = -20°, θ3 = 20°, and θ4 = 40°, respectively, and 50 time samples are assumed. The 5th and 10th sensors of the array are calibrated incorrectly. Based on the incident signal sampling rules, establish the matrix system under the condition of sensor calibration errors:

[0064] Y = B·S + E + N

[0065] Where matrix Y is a 15×50 array sampling matrix, matrix B is a 15×4 array manifold matrix, and each column of the matrix is ​​the steering vector corresponding to the direction of incoming wave, expressed as:

[0066]

[0067] Where d1, d2, ..., d M Let be the distances between the m-th sensor and the first sensor, where m = 1, 2, ..., M. Let E be a 15×50 calibration error matrix and N be a 15×50 additive white Gaussian noise matrix with a signal-to-noise ratio of 20dB.

[0068] In a specific application embodiment, the detailed steps of S2 are as follows:

[0069] S201: Construct dictionary A according to the rule of 1° spatial sampling interval, where each column of A is the steering vector of the incoming wave direction corresponding to the sampling point, and the expression is:

[0070]

[0071] Thus, a matrix system under sparse representation is obtained:

[0072] Y = A·X + E + N

[0073] Here, matrix A is a 15×181 dictionary, and matrix X is a 181×0 sparse signal representation. Both matrices X and E are row-sparse matrices, and this row sparsity property guarantees their solution.

[0074] In a specific application embodiment, the detailed steps of S3 are as follows:

[0075] S301: Based on the matrix system under sparse representation, the following convex optimization formula can be constructed:

[0076]

[0077] Where μ is the smoothing coefficient, λ1 and λ2 are weighting coefficients, and 1 N Let 1 be a column vector of 181 columns, consisting entirely of 1s. M It is a column vector of 15, consisting entirely of 1s.

[0078] In a specific application embodiment, the detailed steps of S4 are as follows:

[0079] S401: Set the smoothing coefficient μ = 10 -4 The tradeoff coefficients λ1 = 8.8 and λ2 = 2, and the maximum number of iterations I. max =1000 and threshold ε=10 -5 .

[0080] S402: Set the initial iteration count i = 0, initialize the solution matrices X = 0 and E = 0, denoted as X 0 and E 0 .

[0081] S403: Calculate the transition matrix P for the i-th iteration. i and Qi The specific calculation method is as follows:

[0082]

[0083] Where ||·||2 represents the vector 2-norm, and diag{·} represents constructing a diagonal matrix using the elements within {}, X i-1 (n) and E i-1 (m) represents the nth row of X and the mth row of E obtained from the (i-1)th iteration update, where n = 1, 2, ..., 181 and m = 1, 2, ..., 15.

[0084] S404: Calculate the solution vector X for the i-th iteration. i and E i The specific calculation method is as follows:

[0085] X i =(A H ·A+λ1P i ) -1 ·A H ·(YE i-1 )

[0086] E i =(I+λ2Q) i ) -1 ·[YA·X i ]

[0087] in(·) H This represents the matrix conjugate transpose operation, (·). -1 This represents the matrix inversion operation.

[0088] S405: Calculate the relative error. The specific calculation method is as follows:

[0089]

[0090] Where f is the objective function, and the specific calculation method is as follows:

[0091]

[0092] After the calculation is complete, update the iteration count i = i + 1.

[0093] S406: Repeat steps S403 to S405 until i is greater than I. max Or the relative error Δ is less than ε.

[0094] In a specific application embodiment, the detailed steps of S5 are as follows:

[0095] S501: The column power diagram obtained by taking the vector 2-norm of each row of the solution matrices X and E is derived from... Figure 3Given. Based on the non-zero row positions of the solution matrices X and E, the corresponding estimated incoming wave directions are -60°, -20°, 20°, and 40°, and the calibration error sensor positions are the 5th and 10th.

[0096] Various aspects of the invention are described in this disclosure with reference to the accompanying drawings, in which numerous illustrative embodiments are shown. The embodiments of this disclosure are not necessarily defined to include all aspects of the invention. It should be understood that the various concepts and embodiments described above, as well as those described in more detail below, can be implemented in any of many ways, because the concepts and embodiments disclosed herein are not limited to any particular implementation. Furthermore, some aspects of the invention disclosed may be used alone or in any suitable combination with other aspects of the invention disclosed.

[0097] While the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the invention. Those skilled in the art can make various modifications and refinements without departing from the spirit and scope of the invention. Therefore, the scope of protection of the present invention shall be determined by the claims.

Claims

1. A method for estimating the direction of arrival of an array sensor under calibration errors, characterized in that: The method includes simultaneously acquiring the direction of incoming wave and the location of the calibrated faulty sensor: S1: Based on the incident signal sampling rules, select the number of time-domain sampling points T, and establish the array sampling matrix system Y = B·S + E + N under the condition of sensor calibration error. Take the number of sampling points T as the column number of the left-hand side matrix. In this matrix, matrix Y is an M×T array sampling matrix, where M is the number of array sensors, matrix B is an M×K array manifold matrix, where K is the number of signal sources, matrix S is a K×T signal matrix, matrix E is an M×T calibration error matrix, and matrix N is an M×T noise matrix. S2: Construct an overcomplete dictionary matrix A of the array manifold matrix according to the preset rules, and obtain the sparse representation Y = A·X + E + N of the array sampling matrix system, where matrix A is an M×N overcomplete dictionary matrix, N is the number of columns in the dictionary, and matrix X is an N×T sparse representation matrix of the signal. S3: Under the low-rank and row sparse decomposition framework, construct a convex optimization formula based on the sparse representation Y = A·X + E + N of the array sampling matrix system; S4: Solve the convex optimization formula using the iterative reweighting method to obtain the solution matrix. and S5: Based on the solution matrix and The non-zero row positions are used to obtain the corresponding incoming wave direction estimates and the positions of the calibrated error sensors.

2. The method for estimating the direction of arrival under array sensor calibration errors according to claim 1, characterized in that: In step S1, the sensor calibration error signal is introduced into the array sampling system model to estimate the direction of incoming waves and obtain the location of the calibrated sensor.

3. The method for estimating the direction of arrival under array sensor calibration errors according to claim 1, characterized in that: In step S2, an overcomplete dictionary matrix A of the array manifold matrix is ​​constructed such that both matrix X and matrix E have row sparsity properties.

4. The method for estimating the direction of arrival under array sensor calibration errors according to claims 1 and 3, characterized in that: Within the low-rank and row-sparse decomposition framework, the convex optimization formula described in step S3 constrains the row sparsity properties of matrices X and E, thereby ensuring that each of the two solution matrices has a unique solution.

5. The method for estimating the direction of arrival under array sensor calibration errors according to claim 1, characterized in that, The iterative reweighting method includes the following steps: S401: Set the smoothing coefficient μ, the tradeoff coefficients λ1 and λ2, and the maximum number of iterations I. max and threshold ε; S402: Initial iteration number i = 0, initialize solution matrices X and E, denoted as X 0 and E 0 ; S403: Calculate the transition matrix P for the i-th iteration. i and Q i The specific calculation method is as follows: Where ||·||2 represents the vector 2-norm, and diag{·} represents constructing a diagonal matrix using the elements within {}, X i-1 (n) and E i-1 (m) represents the nth row of the X matrix and the mth row of the E matrix obtained by the (i-1)th iteration update, where n = 1, 2, ..., N, m = 1, 2, ..., M; S404: Calculate the solution matrix X for the i-th iteration. i and E i The specific calculation method is as follows: X i =(A H ·A+λ1P i ) -1 ·A H ·(Y-E i-1 ) E i =(I+λ2Q i ) -1 ·[Y-A·X i ] in(·) H Represents the matrix conjugate transpose operation, (·) -1 This represents the matrix inversion operation; S405: Calculate the relative error. The specific calculation method is as follows: Where f is the objective function, and the specific calculation method is as follows: After the calculation is complete, update the iteration count i = i + 1; S406: Repeat steps S403 to S405 until the iteration number i is greater than I. max Or the relative error Δ is less than ε.

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