CN116224219A - Array error self-correction atomic norm minimization DOA estimation method - Google Patents

Array error self-correction atomic norm minimization DOA estimation method Download PDF

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CN116224219A
CN116224219A CN202310136558.1A CN202310136558A CN116224219A CN 116224219 A CN116224219 A CN 116224219A CN 202310136558 A CN202310136558 A CN 202310136558A CN 116224219 A CN116224219 A CN 116224219A
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陈涛
申梦雨
郭立民
安琪
胡学晶
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Abstract

An array error self-correction atomic norm minimization DOA estimation method relates to the field of array signal processing. The DOA estimation method aims to solve the problems that the accuracy of DOA estimation cannot be ensured and the calculated amount is small without adding an auxiliary source in the conventional DOA estimation method. The invention comprises the following steps: collecting signal data received by an antenna; acquiring a covariance matrix of signal data received by an antenna, and then performing feature decomposition sequencing on the covariance matrix to obtain noise energy, so as to obtain a covariance matrix without noise errors; acquiring an amplitude error estimated value of the signal receiving array by using a covariance matrix without noise error; constructing an array error self-correction atomic norm minimization model according to the amplitude error estimated value of the signal receiving array, and converting the array error self-correction atomic norm minimization model into a semi-positive definite programming process; and obtaining DOA parameter information of the estimated signal by using an optimal solution of the semi-positive planning process. The method is used for estimating the error of the signal receiving array.

Description

Array error self-correction atomic norm minimization DOA estimation method
Technical Field
The invention relates to the field of array signal processing, in particular to an array error self-correction atomic norm minimization DOA estimation method.
Background
DOA (super-resolution Direction of arrival) is taken as an important branch of the array signal, and DOA estimation technology has important application in radar, communication, sonar and other aspects. The research of DOA estimation algorithm is also widely paid attention to domestic and foreign scholars. The traditional DOA estimation algorithm realizes direction finding of signals by adopting a phase interferometer method and the like. This approach does not allow simultaneous DOA estimation of multiple signals in space. Then, subspace class algorithms represented by MUSIC algorithms solve the super-resolution problem of signals. But this subspace-like approach is less robust at low signal-to-noise ratios and with a small snapshot count. And the sparse reconstruction algorithms such as compressed sensing and the like improve the robustness of the DOA estimation algorithm under the conditions of low signal-to-noise ratio and fewer snapshots. However, the compressed sensing algorithm is affected by grid mismatch, that is, when the angle of the incident signal cannot fall on the grid divided in advance, the estimation performance is reduced. The grid refinement technology can enhance DOA estimation performance, but the method is not completely free from the influence of grid mismatch and can bring larger calculation amount. The DOA estimation by utilizing the atomic norms can solve the problem of grid mismatch in the compressed sensing algorithm, and can obtain better estimated row energy compared with subspace algorithms such as a MUSIC algorithm and the like under the conditions of low signal-to-noise ratio and small block beats. The DOA estimation performance of the classical super-resolution algorithm, the atomic norm minimization and other sparse reconstruction algorithms are all premised on an accurate array flow pattern. However, in an actual array receiving system, the actual array flow pattern is often affected by various factors, so that the array flow pattern of the antenna has deviation, and common array errors include channel amplitude and phase errors, mutual coupling among array elements, array position errors and the like. Therefore, how to obtain an array flow pattern matrix without errors becomes the research focus of researchers at home and abroad.
In the prior art, the array error correction method is realized by measuring, interpolating and storing an array flow pattern in early stage, but the method has larger realization cost and unsatisfactory effect. Then, an active correction method and a self-correction method are provided, wherein the active correction method is to place an auxiliary source with accurately known azimuth information to realize the estimation of the array error parameters, the method has small calculated amount and good estimation performance, but the active correction method requires the direction of the auxiliary signal source to be completely accurate; on the other hand, in practical engineering, the design cost is increased due to the need to place an auxiliary source. The self-correction algorithm realizes joint estimation of signal azimuth and array error, is a high-dimensional nonlinear problem, and therefore, the calculated amount is increased. The current self-correction method has certain requirements on the initial value of the correction matrix, and if the initial value deviates from the actual correction matrix by a large amount, a large estimation error can be brought. Therefore, the current array error correction method also has the problems that the accuracy of DOA estimation cannot be ensured while an auxiliary source is not added, and the calculated amount is large.
Disclosure of Invention
The invention aims to solve the problems that the accuracy of DOA estimation cannot be ensured while an auxiliary source is not added and the calculated amount is large in the conventional DOA estimation method, and provides an array error self-correction atomic norm minimization DOA estimation method.
The array error self-correcting atomic norm minimization DOA estimation method comprises the following specific processes:
step one, collecting signal data received by an antenna;
step two, acquiring a covariance matrix of signal data received by an antenna, and then carrying out feature decomposition sequencing on the covariance matrix to obtain noise energy, so as to obtain a covariance matrix without noise errors;
step three, acquiring an amplitude error estimated value of a signal receiving array by using a covariance matrix without noise error;
step four, constructing an array error self-correction atomic norm minimization model according to the amplitude error estimated value of the signal receiving array obtained in the step three, and converting the array error self-correction atomic norm minimization model into a semi-positive planning process;
the semi-positive programming process has the following formula:
Figure BDA0004085974390000021
Figure BDA0004085974390000022
Figure BDA0004085974390000023
Figure BDA0004085974390000024
wherein tr (·) represents a trace of a solution, toep (u) represents a Toeplitz matrix formed by taking a vector u as a first row, A is equal to or greater than 0, A is a semi-positive matrix A, u is an amount to be optimized including angle information of an incident signal, B is an energy parameter matrix including the incident signal, L is the total number of array elements in a signal receiving array, Z represents a matrix to be optimized, K is a trace of an inverse of an amplitude error estimation matrix, X is signal data received by an antenna,
Figure BDA0004085974390000025
is the inverse of the array error matrix;
and fifthly, acquiring DOA parameter information of antenna received signal data by utilizing an optimal solution of the semi-positive planning process.
Further, the signal data received by the acquisition antenna in the first step is as follows:
X＝ΓAS+N＝WMAS+N
Γ＝WM
wherein Γ is the array error matrix,
Figure BDA0004085974390000026
is composed of L uniform array element groupsA resulting vector matrix of steering vectors, ">
Figure BDA0004085974390000027
θ q For the incidence angle of the q-th signal, τ l Is the time delay of the received signal of the first array element, ω is the angular frequency,/v>
Figure BDA0004085974390000028
Is a signal matrix, T is the total number of snapshots, s q Data representing the q-th signal, +.>
Figure BDA0004085974390000031
Noise matrix, n l For noise data of the i-th array element, Q is the total number of received signals, j is the imaginary unit, w=diag ([ 1, W) 2 ,w 3 ,…,w L ] T ) In the form of an array amplitude error matrix,
Figure BDA0004085974390000032
for the phase error of the array, phi 1l For the phase error between the reference array elements of the first array element, L is more than or equal to 2 and less than or equal to L, w l The amplitude error of the first array element;
the reference array element is the first array element in the signal receiving array;
the signal receiving array is a uniform array with the array element number L, the array element distance is lambda/2, and lambda is the signal wavelength.
Preferably, the method comprises the steps of,
Figure BDA0004085974390000033
wherein ,dl And c represents the speed of light for the position coordinates of the first array element of the array.
Preferably, the method comprises the steps of,
Figure BDA0004085974390000034
further, the step two of obtaining the covariance matrix of the signal data received by the antenna, and then performing feature decomposition sequencing on the covariance matrix to obtain noise energy, thereby obtaining a covariance matrix without noise error, comprising the following steps:
step two, acquiring a covariance matrix of signal data received by an antenna:
Figure BDA0004085974390000035
wherein ,
Figure BDA0004085974390000036
representing noise energy, I L Representing an identity matrix>
Figure BDA0004085974390000037
Is a signal covariance matrix not containing noise term,/->
Figure BDA0004085974390000038
Is the power of the q-th signal, is the conjugate symbol;
secondly, performing eigenvalue decomposition on the covariance matrix of the signal data received by the antenna obtained in the first step to obtain L eigenvalues, performing descending order on the obtained eigenvalues, and then performing weighted average on the L-Q eigenvalues from small to large to obtain noise energy;
and step two, acquiring a covariance matrix without noise errors.
Further, the noise energy is as follows:
Figure BDA0004085974390000039
wherein ,λi Is the ith eigenvalue and has lambda 1 ＞λ 2 ＞…λ Q ＞λ Q+1 ≥…≥λ L 。
Further, the covariance matrix without noise error is expressed as follows:
Figure BDA0004085974390000041
Figure BDA0004085974390000042
Figure BDA0004085974390000043
x(t)＝ΓAs(t)+n(t)
wherein m and n are two different array element labels,
Figure BDA0004085974390000044
r when the snapshot number T is limited X T is the current snapshot number, x (t) is the single snapshot data received by the signal receiving array, s (t) represents the signal data, n (t) is the noise data, [ R ] 0 ] (m,n) For matrix R 0 Elements of the m-th row and n-th column, < >>
Figure BDA0004085974390000045
The phase errors of the m-th and n-th array elements, w m ，w n The amplitude errors of the mth and nth array elements, τ m,q ，τ n,q The time delays for the reception of the q-th signal for the m-th and n-th array elements, respectively.
Further, in the third step, the amplitude error estimated value of the signal receiving array is obtained by using a covariance matrix without noise error, and the following formula is shown:
Figure BDA0004085974390000046
Figure BDA0004085974390000047
wherein R is R 0 R (1) is the first element of vector r, is the dot-division sign,
Figure BDA0004085974390000048
is the energy sum of the Q signals, +.>
Figure BDA0004085974390000049
Is the first array element amplitude error estimated value.
Further, in the fourth step, an array error self-correction atomic norm minimization model is constructed according to the amplitude error estimated value of the signal receiving array obtained in the third step, and the following formula is shown:
Figure BDA00040859743900000410
Figure BDA00040859743900000411
Figure BDA00040859743900000412
where K is the trace of the inverse of the amplitude error estimation matrix,
Figure BDA0004085974390000051
is the inverse of the error matrix.
Further, in the fifth step, the DOA parameter information of the antenna received signal data is obtained by using the optimal solution of the semi-positive programming process, and the method comprises the following steps:
step five, obtaining an optimal solution of the semi-positive planning process, wherein the optimal solution comprises the following formula:
Figure BDA0004085974390000052
step five, two, construction u * Is applied to u by utilizing a vandermonde decomposition method * Processing the Toeplitz matrix to obtain DOA parameter information theta of antenna received signal data;
where θ is the signal incident direction.
The beneficial effects of the invention are as follows:
the method comprises the steps of firstly solving covariance of data received by an array, and then obtaining amplitude error of the array by utilizing the characteristic that covariance matrix main diagonal elements are independent of array phase error and are energy sums of all signals when noise and amplitude error are not considered. And then, the invention utilizes the sparse characteristic of the signal, introduces a constraint term into the classical atomic norm model, and constructs an atomic norm minimization model when array amplitude-phase errors exist. Then, the atomic norm minimization model is utilized to obtain the inverse of the correction error matrix of the channel on the one hand; on the other hand, an optimal solution u containing the information of the incidence angle of the signal is obtained * And then obtaining DOA parameter information by combining with vandermonde decomposition. Compared with the existing amplitude-phase error correction method, the invention does not need to assume to introduce an auxiliary array element without errors, so that the cost in actual engineering use can be reduced. On the other hand, the ANM model is expanded into a model with array errors, and compared with a subspace class algorithm, DOA estimation performance is improved. The atomic norm minimization algorithm is used, so that better robustness can be obtained under the conditions of low signal-to-noise ratio and fewer snapshots; on the other hand, the array error can be estimated simultaneously due to the introduction of constraint terms. Therefore, the invention does not need prior information of the antenna array to estimate the error matrix, and does not need to set an auxiliary array element without errors to obtain the error matrix.
Drawings
FIG. 1 is a diagram of an array model of the present invention;
FIG. 2 is a graph of DOA estimation accuracy at different signal-to-noise ratios for the present invention and for an iterative self-correcting MUSIC algorithm;
FIG. 3 is a graph showing the phase error correction accuracy of different channels under different signal to noise ratios with the iterative self-correction MUSIC algorithm;
FIG. 4 is a graph showing DOA estimation accuracy for different snapshot numbers for the present invention and the iterative self-correcting MUSIC algorithm;
FIG. 5 is a graph showing the phase error correction accuracy of different channels under different snapshot numbers for the iterative self-correction MUSIC algorithm of the present invention;
FIG. 6 is a graph showing DOA estimation accuracy under different phase errors for the present invention and the iterative self-correcting MUSIC algorithm;
fig. 7 is a graph showing the phase error correction accuracy of different channels under different phase errors according to the present invention and the iterative self-correction MUSIC algorithm.
Detailed Description
The first embodiment is as follows: the specific process of the array error self-correction atomic norm minimization DOA estimation method in the embodiment is as follows:
step one, collecting signal data received by an antenna:
as shown in fig. 1, the signal receiving array is a uniform array with an array element number L, the array element spacing is λ/2, and λ is the signal wavelength. The mutual coupling between the array elements is not considered here. The first element of the receiving array is used as a reference element, and the phase errors of other elements are obtained compared with the reference element. Q far-field narrow-band and mutually independent signal models in the space received by the array are as follows:
x(t)＝ΓAs(t)+n(t)
Γ＝WM
wherein x (t) is single snapshot data received by the signal receiving array, s (t) represents signal data, and n (t) is noise data. W=diag ([ 1, W) 2 ,w 3 ,…,w L ] T ) In the form of an array amplitude error matrix,
Figure BDA0004085974390000061
for the phase error of the array, Γ is the array error matrix, φ 1l L is the phase error between the first array element and the first reference array element, L is the index of the array element, L is the total number of the array elements, and w l And L is more than or equal to 2 and less than or equal to L, the amplitude error of the first array element is that j is an imaginary unit, and diag (·) represents a diagonal matrix formed by vectors. Obviously haveW H ＝W * W is true, and there is M H ＝M * H is a conjugate transpose symbol, and x is a conjugate symbol.
The data received by the multi-snapshot array is:
X＝ΓAS+N＝WMAS+N
wherein X is signal data received by an antenna, A is a steering vector matrix, Γ is an array error matrix, and the array error matrix comprises amplitude errors and phase errors.
Figure BDA0004085974390000062
Is a vector matrix composed of L uniform array elements,>
Figure BDA0004085974390000063
can be expressed as: />
Figure BDA0004085974390000064
θ q An incident angle of the q-th signal, d l For the position coordinates of the first element of the array,/->
Figure BDA0004085974390000071
ω is the angular frequency and c is the speed of light. />
Figure BDA0004085974390000072
The time delay of the received signal for the first element. S is a signal matrix, ">
Figure BDA0004085974390000073
T is the number of shots, s q Data representing the q-th signal. />
Figure BDA0004085974390000074
Noise matrix, n l For noise data of the first element, Q is the total number of received signals.
Step two, obtaining a covariance matrix of signal data received by an antenna, and then carrying out feature decomposition sequencing on the covariance matrix to obtain noise energy, thereby obtaining a covariance matrix without noise error, comprising the following steps:
step two, processing signals received by the signal receiving array to obtain a covariance matrix of signal data received by the antenna:
Figure BDA0004085974390000075
wherein ,
Figure BDA0004085974390000076
representing noise energy, I L Representing an identity matrix>
Figure BDA0004085974390000077
R S For a signal covariance matrix not containing noise terms, < +.>
Figure BDA0004085974390000078
Representing the power of the q-th signal.
Step two, performing eigenvalue decomposition on the covariance matrix of the signal data received by the antenna obtained in step two to obtain L eigenvalues, and performing descending order arrangement on the eigenvalues, so that the noise energy can be expressed as a weighted average of L-Q small eigenvalues, namely:
Figure BDA0004085974390000079
wherein ,λi Represents the ith eigenvalue and has lambda 1 ＞λ 2 ＞…λ Q ＞λ Q+1 ≥…≥λ L 。
Step two, acquiring a covariance matrix without noise error:
ideally, the covariance matrix without noise error can be expressed as:
Figure BDA00040859743900000710
in practical applications, since the snapshot number T is limited, the covariance matrix of the actual signal can be obtained by the following formula:
Figure BDA00040859743900000711
wherein T is the current snapshot count, T is the total snapshot count, and at this time, the covariance matrix without noise is:
Figure BDA00040859743900000712
the covariance matrix obtained by the calculation mode has a certain error with the real covariance matrix, so that the array amplitude error obtained by the main diagonal of the covariance matrix also has a certain error with the real amplitude error. When the value of the snapshot number T is larger, the estimated value of the array amplitude error is closer to the actual array amplitude error value.
Specific covariance matrix R 0 The elements of (a) may be expressed as:
Figure BDA0004085974390000081
wherein m and n are two different array element labels, [ R ] 0 ] (m,n) For matrix R 0 The element of the m-th row and n-th column;
Figure BDA0004085974390000082
Figure BDA0004085974390000083
the phase errors of the m-th and n-th array elements, w m ，w n The amplitude errors of the m-th and n-th array elements are respectively; τ m,q ，τ n,q The time delays for the reception of the q-th signal for the m-th and n-th array elements, respectively.
From R 0 As can be seen from the formula of (2), the diagonal elements of the covariance matrixThe phase error of the array is not relevant and the amplitude error of the array can be derived from the main diagonal of the covariance matrix.
Step three, acquiring an amplitude error estimated value of the signal receiving array by using a covariance matrix without noise error:
r is taken 0 The main diagonal element of (2) is denoted as r:
Figure BDA0004085974390000084
/>
wherein ,
Figure BDA0004085974390000085
represents the energy sum, w, of the Q signals l Is the first array element amplitude error estimated value.
The estimated value of the amplitude error of the array, which is obtainable from vector r, is:
Figure BDA0004085974390000086
where r (1) is the first element of the vector r, is a dot-division symbol, and represents each element in the vector r divided by the first element.
Step four, constructing an array error self-correction atomic norm minimization model according to the amplitude error estimated value of the signal receiving array obtained in the step three, and converting the array error self-correction atomic norm minimization model into a semi-positive programming process:
assume that
Figure BDA0004085974390000087
Being the inverse of the array error matrix, the array received data without errors can be expressed as:
Figure BDA0004085974390000088
when noise is not considered, the method comprises>
Figure BDA0004085974390000089
According to the definition of the atomic norms,
Figure BDA0004085974390000091
since the model conforms to the atomic norm, the model conforming to the atomic norm can be constructed by introducing the inverse of the error matrix, and the required optimal solution can be obtained by using the atomic norm minimization method.
Step four, constructing an atomic norm minimization model under the self-correction of array errors, wherein the atomic norm minimization model is as follows:
Figure BDA0004085974390000092
Figure BDA0004085974390000093
where ε is the noise margin and Z represents the matrix to be optimized.
Figure BDA0004085974390000094
For the amplitude error estimation matrix inverse trace, tr (·) represents the matrix-solving trace. />
Figure BDA0004085974390000095
Is the inverse of the error matrix, i.e. +.>
Figure BDA0004085974390000096
And step four, converting the array error self-correction atomic norm minimization model constructed in the step four into a semi-positive definite programming process, wherein the following formula is as follows:
Figure BDA0004085974390000097
Figure BDA0004085974390000098
Figure BDA0004085974390000099
Figure BDA00040859743900000910
where tr (·) represents the trace of the solution, toep (u) represents the Toeplitz matrix formed by taking the vector u as the first column, A.gtoreq.0 represents the semi-positive matrix A, B is the energy parameter matrix containing the incident signal, and u is the amount to be optimized containing the angle information of the incident signal.
Compared with the original atomic norm minimization model, the model provided by the invention can obtain the optimal solution u containing angle information by utilizing the convex optimization method * On the other hand, the inverse estimated value of the array error matrix can be obtained
Figure BDA00040859743900000911
For solving the array error matrix, the error matrix can be obtained by solving +.>
Figure BDA00040859743900000912
The inverse is applied to obtain an error matrix estimate Γ for the array.
Step five, obtaining an optimal solution of the semi-definite programming process, so as to estimate DOA parameter information of the signal:
step five, obtaining an optimal solution of the semi-positive planning process:
Figure BDA00040859743900000913
step five, constructing an optimal solution u * Is used for optimizing a solution u by utilizing a vandermonde decomposition method * Processing the Toeplitz matrix to obtain DOA parameter information theta to be estimated;
where θ is the signal incident direction.
Examples:
to verify the beneficial effects of the invention, the following steps are carried outThe simulation experiment compares the invention with an iterative self-correcting MUSIC algorithm, as shown in figures 2-7, which are named I-MUSIC and C-ANM respectively; fig. 2 is a graph comparing the DOA estimation accuracy of two algorithms at different signal-to-noise ratios, the estimation accuracy being measured by RMSE. The simulation conditions are as follows: the signal to noise ratio was uniformly varied between 0-20dB at 5dB intervals with a snapshot count of 150. The amplitude error is 5.5dB, and the phase error is randomly selected from-40 degrees to 40 degrees. FIG. 3 is a graph comparing the phase error correction accuracy of different channels under different signal-to-noise ratios for two algorithms
Figure BDA0004085974390000101
And (5) measuring. The calculation method comprises the following steps: />
Figure BDA0004085974390000102
Wherein M is the number of Monte Carlo experiments, and L is the number of arrays. />
Figure BDA0004085974390000103
Is the m-th array error estimate. The simulation conditions are the same as those of fig. 2. The array used is a uniform array of 10 elements and figure 3 shows the accuracy curve of the estimated phase error versus the true phase error. Fig. 4 is a graph comparing the DOA estimation accuracy of two algorithms at different snapshot numbers, the estimation accuracy measured in RMSE. The simulation conditions are as follows: the signal-to-noise ratio was 10dB and the snapshot count was 50,100,150,200,300. The amplitude error is 5.5dB, and the phase error is randomly selected from-40 degrees to 40 degrees. FIG. 5 is a graph comparing the phase error correction accuracy curves of different channels under different snapshot numbers for two algorithms>
Figure BDA0004085974390000104
And (5) measuring. The simulation conditions are the same as those of fig. 4. The array used is a uniform array of 10 elements and fig. 5 shows the accuracy curve of the estimated phase error versus the true phase error. Fig. 6 is a graph comparing the DOA estimation accuracy measured in RMSE for two algorithms with different phase errors. The simulation conditions are as follows: the signal-to-noise ratio is 10dB and the number of snapshots is 150. The amplitude error is 5.5dB, and the phase error is thatRandomly selecting from-10 degrees to-10 degrees, -20 degrees to-20 degrees, -30 degrees to-30 degrees, -40 degrees to-40 degrees and-50 degrees to-50 degrees respectively. FIG. 7 is a graph comparing the phase error correction accuracy of different channels under different phase errors for the two algorithms, the accuracy is the same
Figure BDA0004085974390000105
And (5) measuring. The simulation conditions are the same as those of fig. 6. The array used is a uniform array of 10 elements and fig. 7 shows the accuracy curve of the estimated phase error versus the true phase error. Based on comparison of the simulation results, the DOA estimation accuracy is improved while the calculated amount is reduced. />

Claims (10)

1. A method for array error self-correcting atomic norm minimization DOA estimation, the method comprising the steps of:
step one, collecting signal data received by an antenna;
step two, acquiring a covariance matrix of signal data received by an antenna, and then carrying out feature decomposition and sequencing on the covariance matrix to obtain noise energy, so as to obtain a covariance matrix without noise errors;
step three, acquiring an amplitude error estimated value of a signal receiving array by using a covariance matrix without noise error;
step four, constructing an array error self-correction atomic norm minimization model according to the amplitude error estimated value of the signal receiving array obtained in the step three, and converting the array error self-correction atomic norm minimization model into a semi-positive planning process;
the semi-positive programming process has the following formula:
Figure FDA0004085974380000011
Figure FDA0004085974380000012
Figure FDA0004085974380000013
Figure FDA0004085974380000014
wherein tr (·) represents a trace of a solution, toep (u) represents a Toeplitz matrix formed by taking a vector u as a first row, A is equal to or greater than 0, A is a semi-positive matrix A, u is an amount to be optimized including angle information of an incident signal, B is an energy parameter matrix including the incident signal, L is the total number of array elements in a signal receiving array, Z represents a matrix to be optimized, K is a trace of an inverse of an amplitude error estimation matrix, X is signal data received by an antenna,
Figure FDA0004085974380000015
h is the conjugate transpose symbol and ε is the noise margin, which is the inverse of the array error matrix;
and fifthly, acquiring DOA parameter information of antenna received signal data by utilizing an optimal solution of the semi-positive planning process.
2. The array error self-correcting atomic norm minimization DOA estimation method as recited in claim 1, wherein: and (3) collecting signal data received by the antenna in the first step, wherein the signal data are represented by the following formula:
X＝ΓAS+N＝WMAS+N
Γ＝WM
wherein Γ is the array error matrix,
Figure FDA0004085974380000016
is a vector matrix composed of L uniform array elements,>
Figure FDA0004085974380000017
θ q for the incidence angle of the q-th signal, τ l Is the time delay of the received signal of the first array elementLate, ω is angular frequency, ++>
Figure FDA0004085974380000018
Is a signal matrix, T is the total number of snapshots, s q Data representing the q-th signal, +.>
Figure FDA0004085974380000019
Noise matrix, n l For noise data of the i-th array element, Q is the total number of received signals, j is the imaginary unit, w=diag ([ 1, W) 2 ,w 3 ,…,w L ] T ) In the form of an array amplitude error matrix,
Figure FDA0004085974380000021
for the phase error of the array, phi 1l For the phase error between the reference array elements of the first array element, L is more than or equal to 2 and less than or equal to L, w l The amplitude error of the first array element;
the reference array element is the first array element in the signal receiving array;
the signal receiving array is a uniform array with the array element number L, the array element distance is lambda/2, and lambda is the signal wavelength.
3. The array error self-correcting atomic norm minimization DOA estimation method as recited in claim 2, wherein:
Figure FDA0004085974380000022
wherein ,dl And c represents the speed of light for the position coordinates of the first array element of the array.
4. The array error self-correcting atomic norm minimization DOA estimation method as recited in claim 2, wherein:
Figure FDA0004085974380000023
5. a method of array error self-correcting atomic norm minimization DOA estimation according to claim 1, 2, 3 or 4, wherein: the step two of obtaining the covariance matrix of the signal data received by the antenna, then carrying out characteristic decomposition sequencing on the covariance matrix to obtain noise energy, thereby obtaining the covariance matrix without noise error, comprising the following steps:
step two, acquiring a covariance matrix of signal data received by an antenna:
Figure FDA0004085974380000024
wherein ,
Figure FDA0004085974380000025
representing noise energy, I L Representing an identity matrix>
Figure FDA0004085974380000026
Is a signal covariance matrix not containing noise term,/->
Figure FDA0004085974380000027
Is the power of the q-th signal, is the conjugate symbol;
secondly, performing eigenvalue decomposition on the covariance matrix of the signal data received by the antenna obtained in the first step to obtain L eigenvalues, performing descending order on the obtained eigenvalues, and then performing weighted average on the L-Q eigenvalues from small to large to obtain noise energy;
and step two, acquiring a covariance matrix without noise errors.
6. The method for array error self-correcting atomic norm minimization DOA estimation as recited in claim 5, wherein: the noise energy is as follows:
Figure FDA0004085974380000031
wherein ,λi Is the ith eigenvalue and has lambda 1 ＞λ 2 ＞…λ Q ＞λ Q+1 ≥…≥λ L 。
7. The method for array error self-correcting atomic norm minimization DOA estimation as recited in claim 6, wherein: the covariance matrix without noise error is as follows:
Figure FDA0004085974380000032
Figure FDA0004085974380000033
Figure FDA0004085974380000034
x(t)＝ΓAs(t)+n(t)
wherein m and n are two different array element labels,
Figure FDA0004085974380000035
r when the snapshot number T is limited X T is the current snapshot number, x (t) is the single snapshot data received by the signal receiving array, s (t) represents the signal data, n (t) is the noise data, [ R ] 0 ] (m,n) For matrix R 0 Elements of the m-th row and n-th column, < >>
Figure FDA0004085974380000036
The phase errors of the m-th and n-th array elements, w m ，w n The amplitude errors of the mth and nth array elements, τ m,q ，τ n,q The time delays for the reception of the q-th signal for the m-th and n-th array elements, respectively. />
8. The method for array error self-correcting atomic norm minimization DOA estimation as recited in claim 7, wherein: in the third step, the covariance matrix without noise error is used to obtain the amplitude error estimated value of the signal receiving array, and the following formula is adopted:
Figure FDA0004085974380000037
Figure FDA0004085974380000038
wherein R is R 0 R (1) is the first element of vector r, is the dot-division sign,
Figure FDA0004085974380000039
is the energy sum of the Q signals, +.>
Figure FDA00040859743800000310
Is the first array element amplitude error estimated value.
9. The method for array error self-correcting atomic norm minimization DOA estimation as recited in claim 8, wherein: in the fourth step, an array error self-correction atomic norm minimization model is constructed according to the amplitude error estimated value of the signal receiving array obtained in the third step, and the following formula is formed:
Figure FDA0004085974380000041
Figure FDA0004085974380000042
Figure FDA0004085974380000043
Figure FDA0004085974380000044
where K is the trace of the inverse of the amplitude error estimation matrix,
Figure FDA0004085974380000045
is the inverse of the error matrix.
10. The array error self-correcting atomic norm minimization DOA estimation method as recited in claim 9, wherein: in the fifth step, the DOA parameter information of the antenna received signal data is obtained by utilizing the optimal solution of the semi-positive programming process, and the method comprises the following steps:
step five, obtaining an optimal solution of the semi-positive planning process, wherein the optimal solution comprises the following formula:
Figure FDA0004085974380000046
step five, two, construction u * Is applied to u by utilizing a vandermonde decomposition method * Processing the Toeplitz matrix to obtain DOA parameter information theta of antenna received signal data;
where θ is the signal incident direction.
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116699507A (en) * 2023-06-07 2023-09-05 哈尔滨工程大学 Atomic norm-based sparse array complete model error self-correction DOA estimation method
CN117031390A (en) * 2023-08-11 2023-11-10 哈尔滨工程大学 Gridless DOA estimation method based on antenna mutual coupling

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116699507A (en) * 2023-06-07 2023-09-05 哈尔滨工程大学 Atomic norm-based sparse array complete model error self-correction DOA estimation method
CN117031390A (en) * 2023-08-11 2023-11-10 哈尔滨工程大学 Gridless DOA estimation method based on antenna mutual coupling

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