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CN111046591B - Joint estimation method for sensor amplitude-phase error and target arrival angle - Google Patents

Joint estimation method for sensor amplitude-phase error and target arrival angle Download PDF

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CN111046591B
CN111046591B CN201911420375.2A CN201911420375A CN111046591B CN 111046591 B CN111046591 B CN 111046591B CN 201911420375 A CN201911420375 A CN 201911420375A CN 111046591 B CN111046591 B CN 111046591B
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CN111046591A (en
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王伟
董福王
黄平
魏震宇
黄智开
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Harbin Engineering University
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Abstract

The invention provides a joint estimation method of a sensor amplitude-phase error and a target arrival angle, which comprises the steps of establishing a relation between a guide vector and a DoA rough estimation and an off-grid error according to first-order Taylor expansion, constructing a multivariable optimization problem with parameters of the amplitude-phase error and the off-grid error by utilizing orthogonality of a noise subspace and a signal subspace in a multiple signal classification algorithm, and finally realizing error compensation of the sensor and accurate estimation of the DoA through alternative iteration of a sub-problem closed-form solution. The method has the advantages of the super-resolution angle estimation of the MUSIC algorithm while realizing the self-calibration of the sensor, solves the off-network problem in the spectrum searching process, and can greatly improve the accuracy and the robustness of the DoA estimation in practical application.

Description

Joint estimation method for sensor amplitude-phase error and target arrival angle
Technical Field
The invention relates to a joint estimation method of a sensor amplitude-phase error and a target arrival angle, and belongs to the technical field of array signal processing.
Background
The estimation of the target arrival angle based on the antenna array is widely applied to radar, sonar, wireless communication, imaging and other fields. The operating state of the sensors in the array directly affects the accuracy of the angle estimation. In practical applications, the antenna array is always affected by uncertainty factors to generate errors, such as an argument and a phase error, mutual coupling between antennas, and an antenna position error, which directly cause performance degradation and even failure of the conventional angle estimation algorithm. In modern array signal processing techniques, scholars build partially calibrated array models, aiming to jointly estimate the angle of arrival and the amplitude-phase error of the sensor by using data of the same received signal. The existing joint estimation algorithm has the problems of high operation complexity and low accuracy, and the invention can reduce the operation complexity of the algorithm while ensuring the relative high accuracy.
With the development of a compressed sensing technology, the arrival angle estimation algorithm based on a sparse signal reconstruction system shows good performance. The method divides the space into a plurality of grid points to enable the received signals to present space sparsity. However, since the arrival angles are uniformly and continuously distributed in space, an off-grid error is inevitably introduced when the grid is divided, that is, the true target angle is not on the grid point. Similar problems exist in conventional spectrum search algorithms, such as Capon algorithm, MUSIC algorithm. The current solution to this kind of problem is to subdivide the spatial grid minutely, which has the following disadvantages: firstly, the target cannot be guaranteed to be on a discrete grid, and the improvement on the estimation precision is very limited; the second subdivision grid can greatly improve the operation complexity of the algorithm. If amplitude and phase errors possibly existing in a sensor in practical application are considered, an accurate angle estimation cannot be obtained by the existing algorithm.
The traditional DoA estimation is established in an ideal situation, i.e. the error of the sensor possibly existing in the practical application is not considered, so that the performance of the sensor is limited. Meanwhile, under the condition that the known sensor has amplitude and phase errors, the sensor can be subjected to real-time calibration and error compensation at the algorithm level by receiving data through the steps of the method, so that the stability of the angle estimation algorithm is improved, and the application range of the angle estimation algorithm is expanded. Compared with the existing algorithm, the combined estimation method can greatly improve the accuracy and robustness of DoA estimation under the condition of actually existing sensor errors.
Disclosure of Invention
Aiming at the practical problems, the invention provides a joint estimation method of the amplitude and phase errors of the sensor and the target arrival angle by simultaneously considering the amplitude and phase errors of the sensor and the off-grid problem of the traditional spectrum search algorithm, and the joint estimation method is a joint estimation method of the amplitude and phase errors of the sensor and the target arrival angle in a partial calibration array. The method establishes a relation of related data by modeling amplitude-phase errors and off-grid errors and utilizing the orthogonality of a signal subspace and a noise subspace in a subspace algorithm, thereby obtaining a closed solution of error estimation. And finally obtaining accurate angle estimation and amplitude-phase error estimation through alternative iterative calculation of errors.
The purpose of the invention is realized as follows: m omnidirectional receiving antennas which are uniformly distributed form an array, the spacing d of array elements is half of the wavelength lambda, and if a first antenna is taken as a reference array element, the steps are as follows:
the method comprises the following steps: calculating a sample covariance matrix of a received signal in T snapshot times, decomposing an eigenvalue of the covariance matrix to obtain a noise subspace, and obtaining a coarse estimation of the DoA according to a MUSIC spectrum search method;
step two: establishing a signal model with amplitude-phase errors and off-grid errors, and constructing a multivariable optimization problem according to the orthogonality of a signal subspace and a noise subspace;
step three: solving the original problem by using an alternative minimization method, and deducing a closed expression about amplitude-phase error and off-grid error estimation;
step four: setting an initial value of an amplitude-phase error of an uncalibrated sensor, designing a two-layer circulation loop, alternately iterating error parameters in an inner layer until the estimated parameters are unchanged, and sequentially estimating parameters of K targets in an outer layer;
step five: and compensating the off-grid error to obtain accurate DoA estimation, and compensating the amplitude-phase error to finish the self calibration of the sensor.
The invention also comprises the following structural features:
1. the process of constructing the multivariate optimization problem in the second step is as follows:
through the traditional MUSIC spectrum searching mode in the step one, the angles corresponding to K peak values of the spectrum function
Figure BDA0002352222900000021
The DoA is a rough estimation; adopting a first-order Taylor expansion mode, and approximating the guide vector of the real target by using the guide vector of the angle on the grid point, which specifically comprises the following steps:
Figure BDA0002352222900000022
wherein:
Figure BDA0002352222900000023
representing the derivative of the steering vector, betakError between a real target and a grid point;
Figure BDA00023522229000000211
is the steering vector of the array, theta is the space angle; based on the orthogonality relationship between the steering vectors and the noise subspace, the following optimization estimator is designed:
Figure BDA0002352222900000024
Figure BDA0002352222900000025
wherein: gamma denotes the amplitude-phase error vector of the array, Γ (γ) is diag (γ),
Figure BDA0002352222900000026
representing a noise subspace;
Figure BDA0002352222900000027
Figure BDA0002352222900000028
represents McLine, M-McAn all-zero matrix of columns is formed,
Figure BDA0002352222900000029
represents McUnit array of dimensions, McRepresenting the number of calibrated array antennas;
Figure BDA00023522229000000210
represents McAll 1 column vectors of dimensions;
and (3) expanding the objective function according to the definition of the 2 norm, and performing mathematical arrangement according to the following process:
Figure BDA0002352222900000031
wherein processes (a) and (b) satisfy the properties a trace (a) and trace (ab) trace (ba), process (c) being derived according to the following equation:
Figure BDA0002352222900000032
wherein: m and N represent M × M matrices, u represents a vector of M dimensions,
Figure BDA0002352222900000033
representing the Hadamard product, then:
Figure BDA0002352222900000034
Figure BDA0002352222900000035
wherein:
Figure BDA0002352222900000036
2. the closed-form solution process for solving the error estimation in the third step is as follows:
first, the fixed amplitude-phase error γ is unchanged and is converted to βkThe optimal solution is the value of the independent variable when the derivative is zero, and the closed-form solution expression is as follows:
Figure BDA0002352222900000037
second, fix betakIs unchanged and simultaneously has
Figure BDA0002352222900000038
The sub-problem with γ as the argument is therefore:
Figure BDA0002352222900000041
Figure BDA0002352222900000042
wherein:
Figure BDA0002352222900000043
the lagrangian function is constructed as:
Figure BDA0002352222900000044
for lagrange function with respect to gammaHAnd solving the partial derivative to be equal to zero, namely solving the estimation expression of the amplitude-phase error as follows:
Figure BDA0002352222900000045
3. the loop and iteration termination conditions designed in the fourth step are as follows:
in the internal loop, the amplitude-phase error vector gamma is set to 1MAs an initial value, alternately calculating
Figure BDA0002352222900000046
And
Figure BDA0002352222900000047
until the estimated parameters are unchanged, i.e.
Figure BDA0002352222900000048
When the time is up, the calculation is terminated,
Figure BDA0002352222900000049
representing an off-grid error value of the kth target calculated by the ith iteration;
in the outer loop, according to the irrelevance of the incoming wave signals, the estimation of each angle is independent of the calculation of other angle estimation; successively solving off-grid errors of all targets from 1 to K cycle
Figure BDA00023522229000000410
Sum-amplitude-phase error net error
Figure BDA00023522229000000411
The final amplitude-phase error estimate can be expressed as:
Figure BDA00023522229000000412
4. in the fifth step, the accurate DoA is obtained according to the following formula:
Figure BDA00023522229000000413
the steering vector after sensor error compensation is then:
Figure BDA00023522229000000414
compared with the prior art, the invention has the beneficial effects that: 1. the method can estimate the target arrival angle and the array amplitude-phase error by using the same received data, thereby completing the self-calibration of the sensor on the algorithm level. 2. Compared with the traditional algorithm for processing array calibration, the method has higher estimation precision, and can simultaneously estimate the off-grid error of the algorithm for the spectrum search algorithm. 3. Compared with the existing algorithm for estimating the off-grid error, the method has the advantages of high estimation precision and low calculation complexity.
Drawings
FIG. 1 is a block diagram of the overall structure of the present invention;
FIG. 2 is a graph comparing the angle estimation performance of the present invention with the prior art;
FIG. 3 is the variation curve of the root mean square error of the angle estimation with the SNR under the condition of the search step size of 3 deg. according to the present invention;
FIG. 4 is a plot of the root mean square error of the angle estimate as a function of fast beat number for a search step of 3 according to the present invention;
FIG. 5 is a plot of the root mean square error of the amplitude-phase error estimate of the present invention as a function of signal-to-noise ratio;
FIG. 6 is a plot of the root mean square error of the amplitude-phase error estimate versus fast beat number for the present invention;
FIG. 7 is a graph of convergence with increasing number of iterations for the angle estimation RMS error of the present invention at different SNR's;
FIG. 8 is a plot of the angle estimate and the root mean square error of the amplitude and phase of the present invention for convergence as the number of iterations increases.
Detailed Description
The invention is described in further detail below with reference to the drawings and the detailed description.
The first embodiment is as follows: the invention discloses a method for jointly estimating the amplitude-phase error and the target arrival angle of a sensor in a partially calibrated array, which comprises the following steps of:
(1) m omnidirectional receiving antennas which are uniformly distributed form an array, the distance d between array elements is half of the wavelength lambda, and a first antenna is assumed as a reference array element.
(2) And calculating a sample covariance matrix of the received signals in T snapshot times, decomposing the eigenvalue of the covariance matrix to obtain a noise subspace, and obtaining a coarse estimation of the DoA according to a MUSIC spectrum search method.
(3) And establishing a signal model with amplitude and phase errors and off-grid errors, and constructing a multivariable optimization problem according to the orthogonality of a signal subspace and a noise subspace.
(4) And solving the original problem by using an alternative minimization method, and deducing a closed expression about the estimation of the amplitude-phase error and the off-grid error.
(5) Setting an initial value of the amplitude-phase error of an uncalibrated sensor, designing two layers of circulation loops, alternately iterating error parameters in the inner layer until the estimated parameters are unchanged, and sequentially estimating the parameters of K targets in the outer layer.
(6) And compensating the off-grid error to obtain accurate DoA estimation, and compensating the amplitude-phase error to finish the self calibration of the sensor.
The process of constructing the multivariate optimization problem in step (3) is as follows:
through the traditional MUSIC spectrum searching mode in the step (2), the corresponding angles of K peak values of the spectrum function
Figure BDA0002352222900000051
I.e. a coarse estimate of the DoA. However, the accuracy of the algorithm is greatly reduced due to the existence of the amplitude-phase error of the sensor and the off-grid error of the spectrum search. In order to effectively compensate the off-grid error, a first-order Taylor expansion mode is adopted, the guide vector of the angle on the grid point is used for approximating the guide vector of the real target, and the specific formula is as follows:
Figure BDA0002352222900000052
wherein:
Figure BDA0002352222900000061
representing the derivative, beta, of the steering vectorkIs the error between the real target and the grid point.
Figure BDA00023522229000000612
Theta is the steering vector of the array and theta is the spatial angle. From the orthogonality relationship between the steering vectors and the noise subspace, the following optimization estimator can be designed:
Figure BDA0002352222900000062
wherein: γ represents the amplitude-phase error vector of the array, Γ (γ) ═ diag (γ),
Figure BDA0002352222900000063
representing a noise subspace.
Figure BDA0002352222900000064
Figure BDA0002352222900000065
Represents McLine, M-McAll-zero matrix of columns，
Figure BDA0002352222900000066
Represents McUnit array of dimensions, McRepresenting the number of calibrated array antennas;
Figure BDA0002352222900000067
represents McAll 1 column vectors of dimensions.
And expanding the objective function according to the definition of the 2 norm, and performing mathematical arrangement according to the following process:
Figure BDA0002352222900000068
wherein: processes (a) and (b) satisfy the properties a trace (a) and trace (ab) trace (ba), process (c) being derived according to the following equation:
Figure BDA0002352222900000069
wherein: m and N represent M × M matrices, u represents a vector of M dimensions,
Figure BDA00023522229000000610
representing the hadamard product. Therefore, the optimization problem (2) can be equivalently converted into the following form:
Figure BDA00023522229000000611
wherein
Figure BDA0002352222900000071
Figure BDA0002352222900000072
Figure BDA0002352222900000073
The closed-form solution process for solving the error estimation in the step (4) is as follows:
problem (4) is a multivariate optimization problem that can be solved using an alternating minimization algorithm. First, the fixed amplitude-phase error γ is constant, and the original problem can be converted into one about βkThe optimal solution of the problem is the value of the independent variable when the derivative is zero, and the closed-form solution expression is as follows:
Figure BDA0002352222900000074
next, β is fixedkUnchanged while paying attention to
Figure BDA0002352222900000075
Thus a sub-problem with γ as argument can be written as:
Figure BDA0002352222900000076
wherein
Figure BDA0002352222900000077
The problem (6) is a quadratic optimization problem with linear constraints, and the optimal solution of the problem can be solved by using a Lagrange extremum method. The lagrange function is constructed as follows:
Figure BDA0002352222900000078
for function (7) with respect to gammaHThe estimated expression of the amplitude-phase error can be obtained by solving the partial derivative to be equal to zero:
Figure BDA0002352222900000079
the loop and iteration termination conditions designed in the step (5) are as follows:
in the internal loop, the amplitude-phase error vector gamma is set to 1MFor the initial value, equations (5) and (8) are alternately calculated until the estimated parameters are unchanged, i.e.
Figure BDA00023522229000000710
When the time is up, the calculation is terminated,
Figure BDA00023522229000000711
representing the off-grid error value for the kth target calculated for the ith iteration.
In the outer loop, the estimation of each angle is independent of the calculation of the other angle estimates, based on the irrelevancy of the incoming wave signal. Thus, the off-grid errors of all targets are successively found from 1 to K cycles
Figure BDA00023522229000000712
Net error of sum amplitude and phase error
Figure BDA00023522229000000713
The final magnitude-phase error estimate can be expressed as:
Figure BDA00023522229000000714
the accurate DoA in the step (6) is obtained in the following manner:
Figure BDA0002352222900000081
the steering vector after sensor error compensation is:
Figure BDA0002352222900000082
example two:
the target arrival angle and sensor amplitude-phase error estimation method mainly comprises the following steps:
the method comprises the following steps: receiving signal data in T snapshot times by using a sensor, and calculating a corresponding covariance matrix:
Figure BDA0002352222900000083
where y (t) represents the array received signal vector at the t-th snapshot.
Step two: performing eigenvalue decomposition on the obtained covariance matrix to obtain a signal subspace
Figure BDA0002352222900000084
And constructing a MUSIC spectrum function by utilizing the orthogonality of the signal subspace and the noise subspace:
Figure BDA0002352222900000085
in the formula
Figure BDA00023522229000000814
Theta is the steering vector of the array and theta is the spatial angle. Through to the whole space [ -90 DEG, 90 DEG ]]Dividing a discrete grid thetal∈[θ1,θ2,…,θL]Searching K peak values in the formula (13) to obtain coarse estimates of K target angles
Figure BDA0002352222900000086
Step three: calculating the off-grid error of the kth target by:
Figure BDA0002352222900000087
where γ represents the magnitude-phase error vector of the array, the unit array is taken at the first iteration, i.e., assuming no magnitude-phase error is present. Obtaining an estimate of off-grid error
Figure BDA0002352222900000088
The magnitude-phase error of the array is then calculated by:
Figure BDA0002352222900000089
each symbol in formula (14) and formula (15) is as follows:
Figure BDA00023522229000000810
Figure BDA00023522229000000811
Figure BDA00023522229000000812
Figure BDA00023522229000000813
wherein: b (θ) represents a derivative vector of the steering vector a (θ);
Figure BDA0002352222900000091
Figure BDA0002352222900000092
represents McLine, M-McAn all-zero matrix of columns is formed,
Figure BDA0002352222900000093
represents McUnit array of dimensions, McRepresenting the number of calibrated array antennas;
Figure BDA0002352222900000094
represents McAll 1 column vectors of dimensions. Alternately calculating equations (14) and (15) until the estimated parameters are unchanged, i.e.
Figure BDA0002352222900000095
Time-lapse computing to obtain the off-grid error estimate of the kth target
Figure BDA0002352222900000096
Sum array magnitude-phase error estimation
Figure BDA0002352222900000097
Step four: and repeating the third step from the target 1 to the target K to respectively obtain error estimation of all K targets.
Step five: accurate DoA estimation and array amplitude-phase error compensation are obtained as follows:
Figure BDA0002352222900000098
the present invention can also be expressed as follows:
step one, partially calibrating a received signal model of an array
The magnitude-phase error vector of the array is represented by gamma, assuming the first M of the arraycThe remaining M-M of the array elements which are calibratedcIf there is an amplitude-phase error in each array element, then
Figure BDA0002352222900000099
ρmAnd
Figure BDA00023522229000000910
representing the amplitude and phase error of the m-th array element, respectively, the received signal of all K far-field signals arriving at the array can be represented as:
Figure BDA00023522229000000911
wherein Γ (γ) ═ diag (γ), s (t) ═ s1(t),s2(t),…,sK(t)]TIs a complex-valued signal vector, n (t) represents the zero mean, and the variance is σ2Additive complex Gaussian white ofThe noise is generated by the noise-generating device,
Figure BDA00023522229000000912
a manifold matrix of the array is represented,
Figure BDA00023522229000000916
as steering vectors of the array, θkRepresenting the space angle of the target, with the range of [ -90 DEG, 90 DEG ]]。
Step two, decomposing the eigenvalue of the covariance matrix
Figure BDA00023522229000000913
As can be seen from equation (18), the space spanned by the steering vectors is the same as the signal subspace after eigenvalue decomposition
Figure BDA00023522229000000914
That is, the noise subspace
Figure BDA00023522229000000915
Are orthogonal. From this characteristic, a spectral function (13) can be constructed, which results in a coarse estimate of the DoA by maximum search.
Step three, derivation and convergence analysis of error parameter estimation expression
In the second step, the traditional MUSIC algorithm is executed, however, the accuracy of the algorithm is greatly reduced due to the existence of the sensor amplitude and phase error and the spectrum searching off-grid error. In order to effectively compensate the off-grid error, a first-order Taylor expansion mode is adopted, the guide vector of the angle on the grid point is used for approximating the guide vector of the real target, and the specific formula is as follows:
Figure BDA0002352222900000101
wherein
Figure BDA0002352222900000102
Representing the derivative of the steering vector, betakIs the error between the real target and the grid point. From the orthogonality relationship between the steering vectors and the noise subspace, the following optimization estimators can be designed:
Figure BDA0002352222900000103
wherein
Figure BDA0002352222900000104
Figure BDA0002352222900000105
Represents McLine, M-McAn all-zero matrix of columns is formed,
Figure BDA0002352222900000106
represents McUnit array of dimensions, McRepresenting the number of calibrated array antennas;
Figure BDA0002352222900000107
represents McA full 1 column vector of dimensions.
And expanding the objective function according to the definition of the 2 norm, and performing mathematical arrangement according to the following process:
Figure BDA0002352222900000108
wherein processes (a) and (b) satisfy the properties a trace (a) and trace (ab) trace (ba), the derivation of process (c) being according to the following equation:
Figure BDA0002352222900000109
wherein: m and N represent M × M matrices, u represents a vector of M dimensions,
Figure BDA00023522229000001010
representing the hadamard product. Therefore, the number of the first and second electrodes is increased,the optimization problem (20) can be equivalently transformed into the following form:
Figure BDA00023522229000001011
wherein:
Figure BDA0002352222900000111
the problem (22) is a multivariate optimization problem that can be solved using an alternating minimization algorithm. First, the fixed amplitude-phase error γ is constant, and the original problem can be converted into one about βkThe optimal solution of the problem is the value of the independent variable when the derivative is zero, and the closed-form solution expression is as follows:
Figure BDA0002352222900000112
next, β is fixedkUnchanged while paying attention to
Figure BDA0002352222900000113
Thus a sub-problem with γ as argument can be written as:
Figure BDA0002352222900000114
wherein
Figure BDA0002352222900000115
The problem (20) is a quadratic optimization problem with linear constraints, which can be solved for the optimal solution of the problem using the lagrange extremum method. The lagrange function is constructed as follows:
Figure BDA0002352222900000116
the pair function (25) is related to gammaHThe offset derivative is calculated to be equal to zero, so that the estimation of the amplitude-phase error can be obtainedThe expression is calculated as follows:
Figure BDA0002352222900000117
next, equations (23) and (26) are alternately calculated until the iterations converge, i.e.
Figure BDA0002352222900000118
And the process is terminated when the process is finished,
Figure BDA0002352222900000119
representing the off-grid error value for the kth target calculated for the ith iteration.
The existence of the matrix Q in the formula (26)kI.e. the matrix needs to be satisfied is non-singular. In practical calculations to avoid problems due to the matrix QkThe performance degradation caused by approaching the singular matrix needs to adopt a diagonal additional element method. Ready to use
Figure BDA00023522229000001110
To replace QkWhere δ is a very small value, which guarantees the non-singularity of the matrix without affecting the performance.
The convergence problem of the iteration steps of the present invention is discussed below, noting that the objective function of the optimization problem (22) is non-negative, while, due to the acquisition of the optimal closed-form solution, it is guaranteed that the function value is monotonically non-increasing in each iteration step, i.e.,
Figure BDA00023522229000001111
Figure BDA00023522229000001112
and (3) representing the parameter estimation value in the ith iteration, and obviously ensuring the convergence of the iterative algorithm by combining the two properties. Meanwhile, each iterative subproblem can obtain a closed-form solution, and compared with a general numerical algorithm, the method greatly reduces the complexity of operation.
And step four, setting external circulation to sequentially obtain angle estimation of all K targets.
According to the independence between the signals, the estimation process of each target does not influence the estimation of other targets. Thus, an outer loop can be set up to estimate all targets in turn, and in the kth loop, the off-grid error can be obtained
Figure BDA0002352222900000121
And amplitude-phase error of the entire array
Figure BDA0002352222900000122
Finally, the average value of all estimated amplitude-phase errors is adopted as a final estimated value:
Figure BDA0002352222900000123
the effectiveness of the invention can be illustrated by the following simulations:
simulation conditions and contents
1 target arrival angle estimation performance of uniform linear array under condition of amplitude-phase error of sensor
Consider a uniform linear array comprising 10 array elements, and the last 5 array elements of the array have amplitude-phase errors, which are:
Figure BDA00023522229000001210
in order to better show the performance of the invention in terms of off-grid angle estimation, the incoming wave directions of three far-field signals are respectively-15.4423 degrees, 0.3846 degrees and 25.5828 degrees, and the normalized signal energy is simultaneously selected and the signal intensities of different incoming wave directions are ensured to be the same.
Figure BDA0002352222900000124
Representing the definition of the signal-to-noise ratio (dB). Furthermore, the root mean square error is used as a metric to evaluate the performance of the DoA estimation, which is defined as
Figure BDA0002352222900000125
P represents a Monte cardExperiments with the experiments performed in this case,
Figure BDA0002352222900000126
denotes the estimation of the kth target in this experiment, θkAre true values. And P is subjected to 1000 Monte Carlo experiments, the signal-to-noise ratio is from-2 dB to 20dB, the interval is 2dB, and the snapshot number T is 500.
2 under the condition that the search step length is 3 degrees, the invention has the relation between the root mean square error of angle estimation and the signal-to-noise ratio
Consider a uniform linear array comprising 10 array elements, and the last 5 array elements of the array have amplitude-phase errors, which are:
Figure BDA00023522229000001211
in order to better show the performance of the invention in terms of off-grid angle estimation, the incoming wave directions of the signals of the three far fields are respectively-15.4423 degrees, 0.3846 degrees and 25.5828 degrees, and the normalized signal energy is simultaneously selected and the signal intensities of different incoming wave directions are ensured to be the same.
Figure BDA0002352222900000127
Representing the definition of the signal-to-noise ratio (dB). Furthermore, the root mean square error is used as a metric to evaluate the performance of the DoA estimation, which is defined as
Figure BDA0002352222900000128
P denotes the monte carlo experiment at this time,
Figure BDA0002352222900000129
denotes the estimation of the kth target in this experiment, θkAre true values. Taking 1000 Monte Carlo experiments for P, wherein the signal-to-noise ratio is from-2 dB to 20dB, the interval is 2dB, and the snapshot number T is 500.
3. Under the condition that the search step length is 3 degrees, the invention relates the root mean square error of angle estimation to the fast beat number
Consider a uniform linear array comprising 10 array elements, and the last 5 array elements of the array have amplitude-phase errors, which are:
Figure BDA00023522229000001310
in order to better show the performance of the invention in terms of off-grid angle estimation, the incoming wave directions of the signals of the three far fields are respectively-15.4423 degrees, 0.3846 degrees and 25.5828 degrees, and the normalized signal energy is simultaneously selected and the signal intensities of different incoming wave directions are ensured to be the same.
Figure BDA0002352222900000131
Representing the definition of the signal-to-noise ratio (dB). Furthermore, the root mean square error is used as a metric to evaluate the performance of the DoA estimation, which is defined as
Figure BDA0002352222900000132
P denotes the monte carlo experiment at this time,
Figure BDA0002352222900000133
denotes the estimate of the kth target in this experiment, θkAre true values. P was taken 1000 monte carlo experiments with fast beat numbers from 50 to 1000, interval 50, and signal to noise ratio SNR 10 dB.
4. The invention relates to the relation between the root mean square error of the amplitude-phase error estimation and the signal-to-noise ratio
Consider a uniform line array comprising 10 array elements, and the last 5 array elements of the array have amplitude and phase errors, which are:
Figure BDA00023522229000001311
in order to better show the performance of the invention in terms of off-grid angle estimation, the incoming wave directions of the signals of the three far fields are respectively-15.4423 degrees, 0.3846 degrees and 25.5828 degrees, and the normalized signal energy is simultaneously selected and the signal intensities of different incoming wave directions are ensured to be the same.
Figure BDA0002352222900000134
Representing the definition of the signal-to-noise ratio (dB). Furthermore, the root mean square error is used as a metric to evaluate the performance of the DoA estimation, which is defined as
Figure BDA0002352222900000135
P denotes the monte carlo experiment at this time,
Figure BDA0002352222900000136
denotes the estimate of the kth target in this experiment, θkAre true values. Taking 1000 Monte Carlo experiments for P, wherein the signal-to-noise ratio is from-2 dB to 20dB, the interval is 2dB, and the snapshot number T is 500.
5. The invention relates to the relation between the root mean square error and the fast beat number of the amplitude and phase error estimation
Consider a uniform linear array comprising 10 array elements, and the last 5 array elements of the array have amplitude-phase errors, which are:
Figure BDA00023522229000001312
in order to better show the performance of the invention in terms of off-grid angle estimation, the incoming wave directions of the signals of the three far fields are respectively-15.4423 degrees, 0.3846 degrees and 25.5828 degrees, and the normalized signal energy is simultaneously selected and the signal intensities of different incoming wave directions are ensured to be the same.
Figure BDA0002352222900000137
Representing the way in which the signal-to-noise ratio (dB) is defined. Furthermore, the root mean square error is used as a metric to evaluate the performance of the DoA estimation, which is defined as
Figure BDA0002352222900000138
P denotes the monte carlo experiment at this time,
Figure BDA0002352222900000139
denotes the estimation of the kth target in this experiment, θkAre true values. P was taken 1000 monte carlo experiments with fast beat numbers from 50 to 1000, interval 50, and signal to noise ratio SNR 10 dB.
6. The angle estimation method can estimate the convergence condition of the root mean square error along with the increment of the iteration number under different signal-to-noise ratios.
Consider a uniform linear array comprising 10 array elements, and the last 5 array elements of the array have amplitude-phase errors, which are:
Figure BDA0002352222900000147
in order to better show the performance of the invention in terms of off-grid angle estimation, the incoming wave directions of the signals of the three far fields are respectively-15.4423 degrees, 0.3846 degrees and 25.5828 degrees, and the normalized signal energy is simultaneously selected and the signal intensities of different incoming wave directions are ensured to be the same.
Figure BDA0002352222900000141
Representing the way in which the signal-to-noise ratio (dB) is defined. Furthermore, the root mean square error is used as a metric to evaluate the performance of the DoA estimation, which is defined as
Figure BDA0002352222900000142
P denotes the monte carlo experiment at this time,
Figure BDA0002352222900000143
denotes the estimation of the kth target in this experiment, θkAre true values. Taking 1000 Monte Carlo experiments for P, setting the snapshot number T as 500 and the iteration number from 0 to 300.
7. The angle estimation and the root mean square error of the amplitude and the phase of the invention are the convergence condition along with the increment of the iteration number.
Consider a uniform linear array comprising 10 array elements, and the last 5 array elements of the array have amplitude-phase errors, which are:
Figure BDA0002352222900000148
in order to better show the performance of the invention in terms of off-grid angle estimation, the incoming wave directions of three far-field signals are respectively-15.4423 degrees, 0.3846 degrees and 25.5828 degrees, and the normalized signal energy is simultaneously selected and the signal intensities of different incoming wave directions are ensured to be the same.
Figure BDA0002352222900000144
Representing the definition of the signal-to-noise ratio (dB). Furthermore, the root mean square error is used as a metric to evaluate the performance of the DoA estimation, which is defined as
Figure BDA0002352222900000145
P denotes the monte carlo experiment at this time,
Figure BDA0002352222900000146
denotes the estimation of the kth target in this experiment, θkAre true values. Taking 1000 Monte Carlo experiments for P, setting the snapshot number T as 500, setting the SNR as 0dB and setting the iteration number from 0 to 300.
(II) simulation results
1. Target arrival angle estimation performance of uniform linear array under condition of amplitude-phase error of sensor
The comparison of the existing algorithms for solving the off-grid problem is respectively shown in fig. 2, wherein the OGSBI algorithm refers to a sparse bayesian inference algorithm, SUREIR refers to an iterative reweigh algorithm, and MUSIC is an algorithm for accurately dividing grids. It is apparent from fig. 2 that the present invention addresses both amplitude and phase errors and off-grid problems due to the existing algorithms for handling off-grid problems, since these algorithms suffer from a substantial degradation or even failure in the presence of sensor amplitude and phase errors.
2. Under the condition that the search step length is 3 degrees, the invention estimates the relation between the root mean square error of the angle and the signal-to-noise ratio
Fig. 3 shows that in the case of a search step size of 3 °, the angle estimation accuracy of the present invention can be improved as the signal-to-noise ratio increases, and the improvement is increased linearly. For the comparison requirement, the traditional MUSIC algorithm and the MUSIC algorithm with known amplitude and phase errors are added in the simulation, and as can be seen from the graph, even under the condition of the known amplitude and phase errors, if the search step size is too large, the accuracy of the original MUSIC algorithm is reduced. The invention can improve the estimation performance under a larger search step.
3. Under the condition that the search step length is 3 degrees, the invention relates the root mean square error of angle estimation to the fast beat number
Fig. 4 illustrates that the root mean square error of the angle estimate can improve as the number of fast beats increases, but the improvement is not linear. Meanwhile, it can be found that the amplitude-phase error of the array plays a major role in the estimation performance when the step size is small, so that the performance of the conventional MUSIC can be improved under the condition of knowing the amplitude-phase error. But in general, the invention greatly improves the performance of the traditional MUSIC algorithm, especially in the presence of amplitude-phase errors.
4. The invention relates to the relation between the root mean square error of the amplitude-phase error estimation and the signal-to-noise ratio
Fig. 5 shows the estimation of the sensor amplitude and phase errors by the present invention, and also shows the cramer-perot (CRB) boundary as the standard for performance description, it can be seen that the estimation performance of the present invention for the amplitude and phase errors is improved linearly with the increase of the signal-to-noise ratio.
5. The invention relates to the relation between the root mean square error and the fast beat number of the amplitude and phase error estimation
Fig. 6 shows the estimation of the sensor amplitude and phase errors by the present invention, and also shows the cramer-perot (CRB) boundary as the standard for performance description, it can be seen that the estimation performance of the present invention for amplitude and phase errors can be improved as the number of fast beats increases.
6. The angle estimation method has the advantages that the root mean square error is subjected to convergence along with the increment of the iteration number under different signal-to-noise ratios.
Fig. 7 illustrates the iterative convergence of the algorithm under different snr, which can effectively prove the iterative convergence of the invention, especially under high snr, the invention can converge to the DoA estimation level with higher accuracy. Meanwhile, as can be seen from the figure, the convergence of the invention can be ensured no matter what the signal-to-noise ratio is.
7. The angle estimation, amplitude and phase root-mean-square error of the invention is the convergence condition along with the increment of the iteration number.
Fig. 8 shows the convergence of different estimation parameters, i.e. angle, amplitude, phase, with increasing number of iterations. As is apparent from the figure, all the parameters estimated by the present invention can ensure the convergence.
In summary, the present invention provides a joint estimation method for partially calibrating a sensor amplitude-phase error and a target angle of arrival (DoA) in an array, and mainly aims to solve the problem of a rapid decrease in DoA estimation accuracy caused by a sensor error and off-grid search in practical applications. Firstly, establishing a relation between a steering vector and DoA rough estimation and off-grid errors according to first-order Taylor expansion, constructing a multivariable optimization problem with parameters of amplitude-phase errors and off-grid errors by utilizing orthogonality of a noise subspace and a signal subspace in a multi-signal classification algorithm (MUSIC), and finally realizing error compensation of a sensor and accurate estimation of the DoA through alternative iteration of sub-problem closed-form solutions. The detailed process comprises the following steps: and calculating a sample covariance matrix according to the received signals of the antenna array elements, and calculating a noise subspace and a coarse estimation of DoA by using a MUSIC algorithm. Setting an initial value of the amplitude-phase error, and alternately calculating the off-grid error and the amplitude-phase error in the current step until iteration convergence. And finally, obtaining accurate DoA estimation according to the DoA rough estimation and the off-grid error estimation of each target, and compensating the uncalibrated array according to the amplitude-phase error estimation. The method has the advantages of the super-resolution angle estimation of the MUSIC algorithm while realizing the self-calibration of the sensor, solves the off-network problem in the spectrum searching process, and can greatly improve the accuracy and the robustness of the DoA estimation in practical application.

Claims (4)

1. The joint estimation method of the sensor amplitude-phase error and the target arrival angle is characterized by comprising the following steps: m omnidirectional receiving antennas which are uniformly distributed form an array, the spacing d of array elements is half of the wavelength lambda, and if a first antenna is taken as a reference array element, the method comprises the following steps:
the method comprises the following steps: calculating a sample covariance matrix of a received signal in T snapshot times, decomposing an eigenvalue of the covariance matrix to obtain a noise subspace, and obtaining a coarse estimation of the DoA according to a MUSIC spectrum search method;
step two: establishing a signal model with amplitude-phase errors and off-grid errors, and constructing a multivariable optimization problem according to the orthogonality of a signal subspace and a noise subspace;
through the traditional MUSIC spectrum searching mode in the step one, the angles corresponding to K peak values of the spectrum function
Figure FDA0003635998170000011
I.e. a coarse estimation of the DoA; adopt aThe taylor expansion method adopts a grid point angle guide vector to approximate a guide vector of a real target, and specifically comprises the following steps:
Figure FDA0003635998170000012
wherein:
Figure FDA0003635998170000013
representing the derivative, beta, of the steering vectorkError between a real target and a grid point; a (theta) ═ 1, e-j2πd/λsin(θ),…,e-j2π(M-1)d/λsin(θ)]TIs the steering vector of the array, theta is the space angle; based on the orthogonality relationship between the steering vectors and the noise subspace, the following optimization estimator is designed:
Figure FDA0003635998170000014
Figure FDA0003635998170000015
wherein: gamma denotes the amplitude-phase error vector of the array, Γ (γ) is diag (γ),
Figure FDA0003635998170000016
representing a noise subspace;
Figure FDA0003635998170000017
represents McLine, M-McAn all-zero matrix of columns is formed,
Figure FDA0003635998170000018
represents McUnit array of dimensions, McRepresenting the number of calibrated array antennas;
Figure FDA0003635998170000019
represents McAll 1 column vectors of dimensions;
and expanding the objective function according to the definition of the 2 norm, and performing mathematical arrangement according to the following process:
Figure FDA0003635998170000021
wherein processes (a) and (b) satisfy the properties a trace (a) and trace (ab) trace (ba), process (c) being derived according to the following equation:
Figure FDA0003635998170000022
wherein: m and N represent M × M matrices, u represents a vector of M dimensions,
Figure FDA0003635998170000023
representing the Hadamard product, then:
Figure FDA0003635998170000024
Figure FDA0003635998170000025
wherein:
Figure FDA0003635998170000026
step three: solving the original problem by using an alternative minimization method, and deducing a closed expression about amplitude-phase error and off-grid error estimation;
step four: setting an initial value of an amplitude-phase error of an uncalibrated sensor, designing a two-layer circulation loop, alternately iterating error parameters in an inner layer until the estimated parameters are unchanged, and sequentially estimating parameters of K targets in an outer layer;
step five: and (5) compensating the off-grid error to obtain accurate DoA estimation, and compensating the amplitude-phase error to finish the self calibration of the sensor.
2. The method of joint estimation of sensor amplitude-phase error and target angle of arrival of claim 1, wherein: the closed-form solution process for solving the error estimation in the third step is as follows:
first, the fixed amplitude-phase error γ is unchanged and is converted to βkThe optimal solution is the value of the independent variable when the derivative is zero, and the closed-form solution expression is as follows:
Figure FDA0003635998170000031
second, fix betakIs not changed and at the same time has
Figure FDA0003635998170000032
The sub-problem with γ as the argument is therefore:
Figure FDA0003635998170000033
Figure FDA0003635998170000034
wherein:
Figure FDA0003635998170000035
the lagrangian function is constructed as:
Figure FDA0003635998170000036
for lagrange function with respect to gammaHAnd solving the partial derivative to be equal to zero, namely solving the estimation expression of the amplitude-phase error as follows:
Figure FDA0003635998170000037
3. the method of claim 2, wherein the method comprises the steps of: the loop and iteration termination conditions designed in the fourth step are as follows:
in the internal loop, the amplitude-phase error vector gamma is set to 1MAs an initial value, alternately calculating
Figure FDA0003635998170000038
And
Figure FDA0003635998170000039
until the estimated parameters are unchanged, i.e.
Figure FDA00036359981700000310
When the time is up, the calculation is terminated,
Figure FDA00036359981700000311
representing an off-grid error value of the kth target calculated by the ith iteration;
in the external circulation, according to the irrelevance of the incoming wave signals, the estimation of each angle is independent of the calculation of other angle estimations; successively solving off-grid errors of all targets from 1 to K cycle
Figure FDA00036359981700000312
Sum-amplitude-phase error net error
Figure FDA00036359981700000313
The final amplitude-phase error estimate can be expressed as:
Figure FDA00036359981700000314
4. the method of claim 3, wherein the method comprises the steps of: in the fifth step, the accurate DoA is obtained according to the following formula:
Figure FDA00036359981700000315
the steering vector after sensor error compensation is then:
Figure FDA00036359981700000316
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